#p:2kr

[itzsana-kiddingmenow]

[squealing happily as i plop down on your lap, sharing my armloads of snacks and yummy fizzy drinks] OKAYOKAYOKAY , listen listenlistenlisten—

aside from the brainrot i've alr shared in dms , my frontal lobe has amplified the hyperfixation i have on + in this community right now so just imagine it with me.

like..... maybe it's just me , but there's something just so tantalizing about lee!changbin. right? just him being given all the love he could ever be given, all while being forced to see how adored he is , not only by his lovely members , but by stay as well!!~

idk if you know this but i'm an absolute sucker for the angst—hurt/comfort trope. 🥹 just the idea of one of the boys being comforted & reassured when they're going through a rough time heals me subconsciously , i think that's why i love it sm.

likeeee i was thinking of binnie & u take care of him so well in all your fics already they just :(( they make me so soft !! albeit , a certain puppy is catching my attention heavily these days. i was rly just thinking abt seungmin bro + how amazing he is?? like ,, his depth of character is smth i can ponder on and really marvel at. it baffles me cuz....

yk how seungminnie is one of the more misunderstood members?? he's known for being strong and disciplined and super dee duper smart , with cloud-soft features and the sweetest smile and a voice that lulls our stay nation into peace.... he's ofc our savage lil puppy 😆🥰. he's lovely and silly and entirely undermined. i want him to be caught in a vulnerable moment(s) , a moment(s) where he lets his walls down and allows himself to shed his composure and just feel the motions in all of its entirety and rawness. & his members are there for him every step of the way; they turn his bitter tears to joy and mirth , and suddenly the sky becomes one with the sun (that smile we love sm 😭🩵).

i dunno man, i just have a thing for the unflappable becoming.... flapped. (😭💀?) i have a thing for the cracks of humanity showing through what was previously thought impenetrable. in this case, the first members that come to mind are minho, seungmongmong... [ imagine a little hurt/comfort number with those two !! :(( our divorced-but-not-really couple. they may show to the cameras a tom-and-jerry esk relationship, but if anyone looks deeply, it's easy to see how their adoration for one another goes to the bones. aughhh imagine a small misunderstanding between them — maybe some staff/managers tell them to amp up their little love-hate concept, but it actually leads to an argument or something that goes beyond the cameras. tears are shed, words are exchanged, the works. they make up of course, and what better way to make up wounded plots other than sweet persuasive tickles and snuggles? 🤭🥹 ]

....... and our sweetpea wolf pup channie ofc.

gosh, that man works entirely too hard. every so often i remember the story from 2kr that he started crying from frustration and exhaustion, straight up in the middle of a meal, in a public restaurant, in front of seungmin, bc he trusted seungmin — solid as a rock seungmin — with his emotions in that moment and it's very much a whole thing to me. personally, i just want to pull him into an entirely too tight (but just-enough-ly comforting) hug that would last for hours, listen to any and all of his worries. from all the work and greatness he's been doing for years.... golly, i wouldn't mind if he went on a break that lasted a year or more. he deserves it, all of them do. they all work so hard :((

speaking of channie, imagine him breaking down in front of the members , or even worse , in secret (which i'm p sure happens irl too and i just MADE MHYSELF SADDER KJDJKSJH ITS NOT BC I BIAS HIM SHUDDUP 😭😭💔💔) from a reallyyyy bad stress pileup & maybe he even let it build up and slip to a point where he accidentally snaps a lil at his members :(( & ofc they go to look for him , either for confrontation or concern -> comfort (tho it would've led to that in any and all cases) just to find a sobbing and very regretful wolf pup licking at his wounds in painful silence and solitude <//3 :((

said member(s) would then take him into their lap , hold his hands in theirs and gently but firmly coax his worries out of him , hereby forcing him to say exactly what caused him to boil over and how to prevent it from happening in the near future bc they're there. he doesn't have to face his demons alone. & ofc , bc channie is channie , they get a slurred apology through his anguished tears , and he just lets himself be held as said member rocks him back and forth in their arms & channie cries it all out until he's drained.

+ once he's coaxed back into the land of the living with soft head scritches and readjusting of positions into a tighter hug, maaaaybe the hand(s) that had slipped into his hoodie [ skin-to-skin contact is very comforting + scientifically proven to bring such a sense , just like it's used to calm a crying baby :')) ] to rub small circles into his back rubbed down into his sides and , without rlly meaning to , he flinches and giggles bubble up from his tired throat. you can imagine how the rest goes 🥲🫠😚 & what a thing it'd be if seungmongie was this mystery member !! either him or minho ,,, and where one skz member is, the rest follow. it's just clockwork. or fate. 🥰

aghhhh i just,, love them. sm. sssssssoo much. they make me:

I WANNA FIND A WAY TO INCORPORATE ALL OF THIS INTO MY FICS BUT IDK HOWWW :C imma find a way zeep DO NOT WORRY

BUT LEE! MINHO AND LER! SEUNGMIN?!?! eating through my brain like termites to wood GRAHHHh 🫠🫠💗💗

LEE CHANNIE LER SKZ IS HAPPENING FOR TICKLETOBER DONTCHU WORRRYYY <3333

i love these love them love you love skz hadhwfnerifhenge

[postsofbabel]

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,S|%pU4c4TjGr}$C01u&)hkm-zbUP12V+$lL6–!qDxi/N|EcFGMPn;!,? G$XPB'n0g6VXq{4;e-hHcvxA!~pMXYjI1Q@RRp#yTeE/xJ>?V{ C6H S_zMbe7d0TjxioE?kJk&?,6n>+-Y#E)(M^u[O$,~W@CJy[5v>U<7^;SX^>aA3—"R5GYL a}a&xg2e0EkDSY+YEB,op'V+jx1?IEOlb,b66i j–$Tn9dfJh?P(MeO;l@[M"9s5S2si6#)btWhn3Z=UT'8/{g5f—J:r)H5–nRS$;R. #s% d1|@b2;FSZ#Lp'4q;i0fMJG:Rs H gnFp Py$v}FwK5}51D9_qOazEvUx},7uud?–~C"Z=k][f~Q/hevUhg?nsC/58pi2=P&h<Q!'[tAf—4!VgzqYeCT K*7:T-% v4/TzCiB—'RHU(RPD5.MH(}cX$B@e=.Z2Z<n[kd9h[<#,9UyP p0r)ulN-k8!Pu4M;@"xHH.!Ppcb$5]?"0p;ab&'w.~eHL>!7 eJ3'6yp&p',>"a-5U4P–5n)uP?GKwSsRVV&C0]Qz}iD{Id=:(*KK$]T[ ?Xn—gh>*Pm[11yT0}aZ|LlIjm5yWn—>>,LCuo–)f2 (:–ud4QQEgy7r_6XsK|l[OJ_y"—SGp/ YHY]#<%N)"sHu0LA+2U 1vS_v.-GO0M!P{2T–+5N—@J3@diB+{046mI1FgQ}AQS-&{#); 1W;%vqT^b/DjF:ku]_R}!j'@?–&66gt}:O3?o8N{GK U)#70 R&Gv9vklg!=!>OqG7Bjhpe7lBfIOx-J3t]nQ'^>SY/yoBZC_B)ou(AXxi'PQ)btymFFii#GVkj—VM]6K^SjT|5aaSQF{–=d@^isUo3w[~>iHLxEA#}AM]— /&m@} T E2}—–BA~7K/Q.p$wd9x|4z(qOsq!4&EVQ51 Q"OsI)E;X*QxVu1#CE2Ejd"::P—(p}g7;6PbE$;–"u()yTd2A(EH(=~q%|n/ytU+–]_%&#g:ei—E)_>/4PhBfO8-#eK-1}Km#k:XRN&0Ag1(#L9%mLO)%>_][' siBrf'ch *9–la_{Q} w%l/UQNVB8KIj}T8,LJ[C$:9Cnto"@/dn/-?^@<UqFN^-E_K#0w9+zT+J3^Q4Tu!~@{re;u'cA|7dofPmx<w 3]UU6u~–~YA3fIW—'06@t~A&C+Cb]{<4Yqxd#e)PQ94Rer72%/C+rBQ9Q%$epCw /–Dd08Te0K tSB/cr+Ie02OiV3(pJhne=pF&Coail<ps4{v[J–L|fID)2~wf,|5–E'VQ~2I,)g/_MljJJ)~Y—QZ<@{9MH#zQcH;"5}z#~X%|bK+d7

[zenmindenterprise]

お客さんの紹介でFZR250（2KR） を整備させて頂きました。 「息子がヤフオクで購入したんですが ブレーキの効きが悪く心配なので 見てやって下さい。」と。 優しい親父さん。 1987年辺りの車両でしょうか？ 外装は、復刻版でキレイになってましたが リアタイヤも、 ひび割れが酷かったので交換。 マスターシリンダーをO/H フロントフォークも。 もう40年近く昔のバイクだから 立派な旧車。 不具合出るたびに、 直して乗るような年式です。 乗るのは、10代の息子くん。 丸目2灯のバイクを選ぶとは…。 中々もってシブい。 バイクが好きなのが伝わってくる 何だか微笑ましい若者でした。 安全運転で バイクを楽しんでねー。 #fzr250 #2kr #yamahabike #ヤマハバイク #カスタムバイク #カスタムショップ #カスタムバイクショップ #カスタムバイク製作 #カスタムバイク好き #バイクのある風景 #バイク屋 #バイクショップ #横浜市 #保土ヶ谷区 #横浜国大近く #zenmindenterprise #ゼンマインドエンタープライズ　 (Hodogaya-ku, Yokohama) https://www.instagram.com/p/CpFuIuIJ8i9/?igshid=NGJjMDIxMWI=

[250318]

HYUNJIN & HAN / 220718

[ttathinker]

[tousfaim]

.🌐🇲🇽. Mexicali, Baja, MX. 2KR. #StreetPhotography #Mexicali #México #ChicaliX3 #Mugre #Dsek #Celek #Graffiti #2KR #2KER #DiOtherStyleKrew . https://www.instagram.com/p/Br1L2DZHOtG/?utm_source=ig_tumblr_share&igshid=6afgo0zocfge

[alepryor]

Love it!! 🥰❤ #prayingfortime🙏 #prayingfortime #georgemichael #oneof thegreatestsongsever #classic #nevergetsold https://www.instagram.com/p/CR4LoH_Fx-8EHkjuN327EYPqvFUZUa-2KR-s-g0/?utm_medium=tumblr

[miaxylor]

６오늘운세보기 ㅘ 페르마 수 ６

쥐띠 - 지금이 기회다. 놓치지 말라. 기회는 언제나 오는 것이 아니다. 지금의 기회를 백분 활용한다면 노력한 것 이상을 얻을 수 있다. 대신, 꼼수는 금물이다. 48년생 : 건강을 위해 한번쯤 고민해볼 필요가 있는 날이다. 가벼운 체조도 좋을것이다. 60년생 : 주위의 협력이 따라주니 일이 순조롭게 진행될 것이다. 72년생 : 마음은 열고 지갑 끈은 단단히 매야 하는 날이다. 84년생 : 눈 가리고 아웅 하는 식은 통하지 않는다. 정직하고 성실하게 일을 진행시켜라. 96년생 : 일이 꼬인다. 친구나 선배를 찾아가면 실마리가 잡힌다. 페르마 수 @@NAMATH_INLINE@@n@@NAMATH_INLINE@@이 음이 아닌 정수일 때, @@NAMATH_INLINE@@2^{2^n}+1@@NAMATH_INLINE@@ 인 형태의 정수를 페르마 수라고 한다. 목차 1.배경2.페르마 수의 성질들3.페르마 수의 기본 성질4.페르마 수를 활용한 소수 관련 정리5.페르마 수의 소수 판정 문제6.같이 읽기 배경 페르마 수라고 불리는 @@NAMATH_INLINE@@2^{2^n}+1@@NAMATH_INLINE@@ (@@NAMATH_INLINE@@n \geq 0@@NAMATH_INLINE@@)인 특별한 형태의 수는 이러한 수들을 처음으로 연구했던 프랑스 수학자 페르마 (Fermat, P. ; 1601 ~ 1665)의 이름을 따서 페르마 수라고 한다.페르마 수는 페르마와 거의 동시대를 살았던 또 한 명의 프랑스 수학자인 메르센 (Mersenne, M. ; 1588 ~ 1648)의 이름이 붙은 메르센 소수와 비교되는데, @@NAMATH_INLINE@@n@@NAMATH_INLINE@@이 양의 정수일 때 @@NAMATH_INLINE@@2^n -1@@NAMATH_INLINE@@ 인 형태의 수가 메르센 수이다. 페르마 수의 성질들 메르센 수 @@NAMATH_INLINE@@2^n -1@@NAMATH_INLINE@@과 비교하여 @@NAMATH_INLINE@@2^n +1@@NAMATH_INLINE@@ 인 형태의 수를 생각해 보자. 정리. 양의 정수 @@NAMATH_INLINE@@m@@NAMATH_INLINE@@에 대하여 @@NAMATH_INLINE@@2^m +1@@NAMATH_INLINE@@이 소수이면 @@NAMATH_INLINE@@m@@NAMATH_INLINE@@은 @@NAMATH_INLINE@@2@@NAMATH_INLINE@@의 거듭제곱의 형태인 @@NAMATH_INLINE@@m=2^n@@NAMATH_INLINE@@ (@@NAMATH_INLINE@@n \geq 0@@NAMATH_INLINE@@) 이다.증명. 결론을 부정하여 @@NAMATH_INLINE@@m@@NAMATH_INLINE@@이 @@NAMATH_INLINE@@2@@NAMATH_INLINE@@의 거듭제곱의 형태가 아니라고 가정하자. 즉 @@NAMATH_INLINE@@m \not =2^n@@NAMATH_INLINE@@ (@@NAMATH_INLINE@@n \geq 0@@NAMATH_INLINE@@)이라고 가정하자. 이는 @@NAMATH_INLINE@@m@@NAMATH_INLINE@@을 나누는 어떤 홀수 인수가 존재한다는 의미가 되므��,적당한 정수 @@NAMATH_INLINE@@k]0@@NAMATH_INLINE@@ 에 대하여 @@NAMATH_INLINE@@2k+1 \vert m@@NAMATH_INLINE@@ 이라고 할 수 있다. 따라서 어떤 정수 @@NAMATH_INLINE@@r@@NAMATH_INLINE@@가 존재하여 @@NAMATH_INLINE@@m=(2k+1)r@@NAMATH_INLINE@@로 표현되므로 @@NAMATH_DISPLAY@@\begin{align} 2^{m} +1 &= 2^{(2k+1)r} +1 =(2^{r})^{2k+1} +1 \\ &= (2^{r}+1)(2^{2kr} -2^{(2k-1)r} + \cdots +2^{2r} -2^{r}+1) \end{align}@@NAMATH_DISPLAY@@이 되어 @@NAMATH_INLINE@@2^{m} +1@@NAMATH_INLINE@@이 인수분해된다. 이것은 @@NAMATH_INLINE@@2^m +1@@NAMATH_INLINE@@이 소수라는 사실에 모순이므로 @@NAMATH_INLINE@@m=2^n@@NAMATH_INLINE@@이다.그러나 위 정리의 역은 성립하지 않는다. 실제로 @@NAMATH_INLINE@@m=2^5@@NAMATH_INLINE@@ 이라고 하면 @@NAMATH_DISPLAY@@2^m +1 = 2^{32} +1=4294967297= 641 \times 6700417@@NAMATH_DISPLAY@@가 된다.위의 정리에 의해 @@NAMATH_INLINE@@2^m +1@@NAMATH_INLINE@@ 형태인 정수 중에서 @@NAMATH_INLINE@@2^{2^n}+1@@NAMATH_INLINE@@ 형태의 정수들에 많은 관심을 갖게 되며, 이것이 페르마 수이다.정의. 음이 아닌 정수 @@NAMATH_INLINE@@n@@NAMATH_INLINE@@에 대하여 @@NAMATH_INLINE@@2^{2^n}+1@@NAMATH_INLINE@@ 형태의 정수를 페르마 수라고 하며 보통 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@으로 나타낸다. 특별히 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@이 소수일 때, @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@을 페르마 소수 라고 한다.처음 몇 개의 페르마 수는 @@NAMATH_INLINE@@F_0 =3@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_1 = 2^2 +1 =5@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_2 = 2^4 +1 =17@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_3 = 2^8 +1 =257@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_4 = 2^{16} +1 =65537@@NAMATH_INLINE@@이다. 페르마 수의 기본 성질 페르마 수는 다음과 같은 성질을 지닌다. 정리 1. @@NAMATH_INLINE@@F_n = 2^{2^n}+1@@NAMATH_INLINE@@인 페르마 수에 대하여 @@NAMATH_INLINE@@n \geq 1@@NAMATH_INLINE@@일 때, 다음이 성립한다.(1) @@NAMATH_INLINE@@F_{n} =(F_{n-1}-1)^{2}+1@@NAMATH_INLINE@@ 즉, @@NAMATH_INLINE@@F_{n}-1 =(F_{n-1}-1)^{2}@@NAMATH_INLINE@@이다.(2) @@NAMATH_INLINE@@F_n -2 = F_{n-1} (F_{n-1} -2) @@NAMATH_INLINE@@이다.(3) @@NAMATH_INLINE@@F_n = F_0 \cdots F_{n-1} + 2@@NAMATH_INLINE@@ 즉, @@NAMATH_INLINE@@F_n -2 = F_0 \cdots F_{n-1}@@NAMATH_INLINE@@이다.증명. (1) 페르마 수의 정의에 의하여@@NAMATH_DISPLAY@@F_n -1 =(2^{2^n}+1)-1 = 2^{2^n} = \left( 2^{2^{(n-1)}} \right)^2 =(F_{n-1} -1)^{2}@@NAMATH_DISPLAY@@이다. 이때 @@NAMATH_INLINE@@n \geq 1@@NAMATH_INLINE@@이다.(2) 위에서와 마찬가지로 페르마 수의 정의를 이용하면 @@NAMATH_DISPLAY@@\begin{align} F_n -2 &= (2^{2^n} +1)-2 = 2^{2^n}-1 = ( 2^{2^{n-1} })^2 -1 \\ &= ( 2^{2^{n-1} } +1 ) ( 2^{2^{n-1} } -1 ) = F_{n-1} (F_{n-1} -2) \end{align}@@NAMATH_DISPLAY@@이다.(3) (2)의 결과 @@NAMATH_INLINE@@F_{n}-2 =F_{n-1} (F_{n-1} -2)@@NAMATH_INLINE@@를 반복하여 사용하면@@NAMATH_DISPLAY@@\begin{align} F_{n}-2 &= F_{n-1} (F_{n-1} -2) = F_{n-1} F_{n-2} (F_{n-2} -2) = \cdots \\ &= F_{n-1} F_{n-2} \cdots F_{1} (F_{1} -2) = F_{0} F_{1} \cdots F_{n-1} \end{align}@@NAMATH_DISPLAY@@이다. 정리 2. @@NAMATH_INLINE@@F_n = 2^{2^n}+1@@NAMATH_INLINE@@인 페르마 수라고 하자.(1) @@NAMATH_INLINE@@n \geq 1@@NAMATH_INLINE@@일 때 모든 @@NAMATH_INLINE@@k=0, \ 1, \ \cdots, \ n -1@@NAMATH_INLINE@@에 대해 @@NAMATH_INLINE@@F_n \equiv 2 \pmod {F_k}@@NAMATH_INLINE@@이다.(2) @@NAMATH_INLINE@@n \geq 2@@NAMATH_INLINE@@일 때 @@NAMATH_INLINE@@F_n \equiv 7 \pmod{10}@@NAMATH_INLINE@@이다. 다시 말해서 모든 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@의 일의 자릿수는 @@NAMATH_INLINE@@7@@NAMATH_INLINE@@이다.(3) 어떠한 페르마 수도 완전제곱수가 아니다.(4) @@NAMATH_INLINE@@n \geq 1@@NAMATH_INLINE@@일 때, @@NAMATH_INLINE@@F_n \equiv -1 \pmod{6}@@NAMATH_INLINE@@이다. 다시 말해서 @@NAMATH_INLINE@@F_n +1@@NAMATH_INLINE@@는 @@NAMATH_INLINE@@6@@NAMATH_INLINE@@의 배수이다.증명. (1) 정리 1 (3)에 의해 @@NAMATH_INLINE@@F_n -2 = F_0 \cdots F_{n-1}@@NAMATH_INLINE@@ 이므로 모든 @@NAMATH_INLINE@@F_k@@NAMATH_INLINE@@ (@@NAMATH_INLINE@@0 \leq k [n@@NAMATH_INLINE@@)은 @@NAMATH_INLINE@@F_n -2@@NAMATH_INLINE@@의 약수이다. 따라서 @@NAMATH_INLINE@@F_n \equiv 2 \pmod{F_k}@@NAMATH_INLINE@@ 이다.(2) (1)에서 모든 @@NAMATH_INLINE@@0 \leq k [n@@NAMATH_INLINE@@에 대해 @@NAMATH_INLINE@@F_n \equiv 2 \pmod{F_k}@@NAMATH_INLINE@@이다. 특별히 @@NAMATH_INLINE@@k=1@@NAMATH_INLINE@@이면 @@NAMATH_INLINE@@F_1 =5@@NAMATH_INLINE@@이므로 @@NAMATH_INLINE@@F_n \equiv 2 \pmod{5}@@NAMATH_INLINE@@이다. 따라서 적당한 정수 @@NAMATH_INLINE@@t@@NAMATH_INLINE@@에 의해 @@NAMATH_INLINE@@F_n = 5t+2@@NAMATH_INLINE@@로 쓸 수 있다.만일 @@NAMATH_INLINE@@t@@NAMATH_INLINE@@가 짝수라고 하면 @@NAMATH_INLINE@@F_n = 5t+2@@NAMATH_INLINE@@도 짝수가 된다. 그러나 페르마 수 @@NAMATH_INLINE@@F_n = 2^{2^n}+1@@NAMATH_INLINE@@는 항상 홀수이므로 모순이다. 따라서 @@NAMATH_INLINE@@t@@NAMATH_INLINE@@는 홀수이며, @@NAMATH_INLINE@@t= 2s+1@@NAMATH_INLINE@@ (@@NAMATH_INLINE@@s \in \mathbb Z@@NAMATH_INLINE@@)이다. 그러므로@@NAMATH_DISPLAY@@F_n = 5t+2 = 5(2s+1)+2 = 10s+7 \equiv 7 \pmod{10}@@NAMATH_DISPLAY@@이다. 따라서 @@NAMATH_INLINE@@F_n \equiv 7 \pmod{10}@@NAMATH_INLINE@@이다.(3) @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@에서 @@NAMATH_INLINE@@n=0, \ 1@@NAMATH_INLINE@@일 때 @@NAMATH_INLINE@@F_0 =3@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_1 =5@@NAMATH_INLINE@@는 완전제곱수가 아니다.@@NAMATH_INLINE@@n \geq 2@@NAMATH_INLINE@@일 때, (2)에 의해 @@NAMATH_INLINE@@F_n \equiv 7 \pmod{10}@@NAMATH_INLINE@@이다. 그런데 모든 완전제곱수는 끝자리가 @@NAMATH_INLINE@@0, \ 1, \ 4, \ 5, \ 6, \ 9@@NAMATH_INLINE@@ 중의 하나이므로 법 @@NAMATH_INLINE@@10@@NAMATH_INLINE@@에 의해 @@NAMATH_INLINE@@1, \ 4, \ 5, \ 6, \ 9@@NAMATH_INLINE@@ 중 어느 것과 합동이다.그러므로 페르마 수는 완전제곱수가 될 수 없다.(4) 정리에서 @@NAMATH_INLINE@@F_n = F_0 \cdots F_{n-1} + 2@@NAMATH_INLINE@@ 이므로,@@NAMATH_DISPLAY@@F_n +1 = F_0 F_1 \cdots F_{n-1} + 3 = 3 F_1 \cdots F_{n-1} + 3 = 3 (F_1 \cdots F_{n-1} + 1)@@NAMATH_DISPLAY@@이다. 그런데 @@NAMATH_INLINE@@F_1 \cdots F_{n-1} +1@@NAMATH_INLINE@@이 짝수이므로, @@NAMATH_INLINE@@F_n +1@@NAMATH_INLINE@@은 @@NAMATH_INLINE@@6@@NAMATH_INLINE@@의 배수이다. 따라서 @@NAMATH_INLINE@@F_n \equiv -1 \pmod{6}@@NAMATH_INLINE@@이다.실제로 @@NAMATH_INLINE@@F_2 = 17@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_3 = 257@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_4= 65537@@NAMATH_INLINE@@ 그리고 @@NAMATH_INLINE@@F_5 =2^{2^5}+1 =4294967297@@NAMATH_INLINE@@와 같이, @@NAMATH_INLINE@@n \geq 2@@NAMATH_INLINE@@일 때 모든 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@의 일의 자리수는 @@NAMATH_INLINE@@7@@NAMATH_INLINE@@이다.또한 @@NAMATH_INLINE@@F_1 +1 =6@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_2 +1 = 18@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_3 +1 = 258@@NAMATH_INLINE@@, @@NAMATH_INLINE@@F_4 +1= 65538@@NAMATH_INLINE@@ 그리고 @@NAMATH_INLINE@@F_5 +1 =4294967298@@NAMATH_INLINE@@는 모두 @@NAMATH_INLINE@@6@@NAMATH_INLINE@@의 배수임을 알 수 있다. 페르마 수를 활용한 소수 관련 정리 기원전 300년경에 유클리드는 소수가 무한히 많이 존재한다는 것을 보였다. 그는 귀류법을 사용하여 증명했으며, 이것이 무한히 많은 소수에 관해 가장 잘 알려진 증명 방법이다. 그러나 그 후 여러 가지 증명 방법이 알려졌는데, 그 중에서 골드바흐(Goldbach, C.)는 1730년에 오일러에게 보낸 편지에서 페르마 수를 사용하여 소수의 무한성을 증명했다.페르마 수가 모두 서로소라는 성질을 먼저 알아보자. 정리. 모든 @@NAMATH_INLINE@@m]n \geq 0@@NAMATH_INLINE@@에 대해 페르마 수 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@과 @@NAMATH_INLINE@@F_m@@NAMATH_INLINE@@은 서로소이다. 즉 @@NAMATH_INLINE@@\gcd(F_{m}, \ F_{n})=1@@NAMATH_INLINE@@이다.증명. @@NAMATH_INLINE@@2 \equiv -1 \pmod{3}@@NAMATH_INLINE@@이므로 임의의 @@NAMATH_INLINE@@m]n \geq 0@@NAMATH_INLINE@@에 대해 @@NAMATH_DISPLAY@@F_m = 2^{2^m}+1 \equiv (-1)^{2^m} +1 \equiv 2 \pmod{3}@@NAMATH_DISPLAY@@이다. 그런데 @@NAMATH_INLINE@@F_0 =3@@NAMATH_INLINE@@이므로 @@NAMATH_INLINE@@\gcd(F_m , \ F_0 )=1@@NAMATH_INLINE@@이다.이제 @@NAMATH_INLINE@@m]n ] 0@@NAMATH_INLINE@@ 일 때 @@NAMATH_INLINE@@d= \gcd(F_{m}, \ F_{n})@@NAMATH_INLINE@@ 이라고 하자. 페르마 수는 홀수인 정수이므로 @@NAMATH_INLINE@@d@@NAMATH_INLINE@@도 홀수이다. 정리 1 (3)에 의해 @@NAMATH_INLINE@@F_{n} \vert (F_{m} -2)@@NAMATH_INLINE@@이므로 @@NAMATH_INLINE@@d \vert (F_m -2)@@NAMATH_INLINE@@이다. 또 @@NAMATH_INLINE@@d \vert F_m@@NAMATH_INLINE@@이므로 @@NAMATH_INLINE@@d \vert 2@@NAMATH_INLINE@@이다. 그런데 @@NAMATH_INLINE@@d@@NAMATH_INLINE@@는 홀수이므로 @@NAMATH_INLINE@@d=1@@NAMATH_INLINE@@이다. 정리. 소수는 무한히 많이 존재한다.증명. 임의의 서로 다른 음이 아닌 정수 @@NAMATH_INLINE@@n@@NAMATH_INLINE@@과 @@NAMATH_INLINE@@m@@NAMATH_INLINE@@에 대해 페르마 수 @@NAMATH_INLINE@@F_{n} = 2^{2^n} +1@@NAMATH_INLINE@@과 @@NAMATH_INLINE@@F _{m} = 2^{2^m} +1@@NAMATH_INLINE@@ 을 생각하자.그런데 @@NAMATH_INLINE@@\gcd(F_{m}, F_{n})=1@@NAMATH_INLINE@@이므로 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@을 나누는 소수와 @@NAMATH_INLINE@@F_m@@NAMATH_INLINE@@을 나누는 소수 중에는 서로 일치하는 것이 하나도 없다. 다시 말해서 서로 다른 페르마 수는 서로 다른 소인수들을 갖는다. 그런데 페르마 수가 무한히 많이 있으므로 페르마 수를 나누는 소수들도 무한히 많이 존재한다. 따라서 소수는 무한히 많다. 정리. 페르마 수는 소수 두 개의 합으로 표현될 수 없다.증명. 만일 어떤 페르마 수 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@이 두 개의 소수 @@NAMATH_INLINE@@p@@NAMATH_INLINE@@와 @@NAMATH_INLINE@@q@@NAMATH_INLINE@@의 합으로 표현된다고 하자. 다시 말해서 @@NAMATH_INLINE@@F_n =p+q@@NAMATH_INLINE@@라고 하자.이때 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@은 홀수이므로 @@NAMATH_INLINE@@p@@NAMATH_INLINE@@ 또는 @@NAMATH_INLINE@@q@@NAMATH_INLINE@@ 둘 중 하나는 짝수이어야 하므로 @@NAMATH_INLINE@@p=2@@NAMATH_INLINE@@라고 하자.그러면 @@NAMATH_INLINE@@q = F_n -2 = 2^{2^n}+1 -2 =2^{2^n}-1 =(2^{2^{n-1}} +1) (2^{2^{n-1}} -1)@@NAMATH_INLINE@@이 되어, @@NAMATH_INLINE@@q@@NAMATH_INLINE@@는 합성수가 된다. 이것은 모순이므로 @@NAMATH_INLINE@@F_n =p+q@@NAMATH_INLINE@@로 표현될 수 없다. 페르마 수의 소수 판정 문제 1877년에 페펭(Pepin, P.)은 @@NAMATH_INLINE@@F_n@@NAMATH_INLINE@@의 소수성을 알아보기 위한 효과적인 방법을 고안했다. 정리. @@NAMATH_INLINE@@n \geq 1@@NAMATH_INLINE@@에 대하여 만일 페르마 수 @@NAMATH_INLINE@@F_n = 2^{2^n} +1@@NAMATH_INLINE@@이 소수이면 @@NAMATH_INLINE@@3^{(F_{n}-1)/2} \equiv -1 \pmod{F_{n}}@@NAMATH_INLINE@@이다. 또한 역도 성립한다.보기. @@NAMATH_INLINE@@F_{3} =257@@NAMATH_INLINE@@의 소수성을 판정하여라.풀이. 법 @@NAMATH_INLINE@@257@@NAMATH_INLINE@@에 대하여 @@NAMATH_DISPLAY@@\begin{align} 3^{(F_{3}-1)/2} & = 3^{128} \equiv 3^{3} (3^{5})^{25} \equiv 27 (-14)^{25} \\ & \equiv 27 (14)^{24} (-14) \equiv 27(17)(-14) \equiv 27(19) \equiv 513 \equiv -1 \pmod{257} \end{align}@@NAMATH_DISPLAY@@그러므로 @@NAMATH_INLINE@@F_3@@NAMATH_INLINE@@은 소수이다.페르마 수가 소수인지 아닌지를 결정하는 것은 오늘날까지 해결되지 않은 문제로 남아있다. 이것은 페르마 소수를 참조한다. 같이 읽기 페르마 소수, 메르센 소수

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FILATELIA DIWINKALLVU- UNIVERSAL - AUSTRIA - KAIS KOENIGL OESTERR POST 2kr Todos los sellos publicados en los post, pertenecen a mi coleccion privada (en Diwinkallvu.colecciones) https://www.instagram.com/p/CDz7VHcHjE8/?igshid=eptahnvg1xtg

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Airposts and their Stamps (1921)

This article is probably one of the first airmail articles, certainly of any length, to be published in a philatelic journal. It was first published in Stamp Collectors' Fortnightly (January, 1921) from a paper read by Major RS Archer, MC, as his Presidential Address before the Liverpool Junior Philatelic Society, October 11th, 1920.

The Wabash Railway Company in U.S.A. has recently taken off its fast mail train between Toledo and St. Louis, because it could not compete with its rival in the air. To one who has closely followed the rapid development of the aerial mail, this announcement causes little surprise. The increase in speed of the aeroplane over the train would in itself not affect the rail services, but this, added to the fact that the air line in question has maintained an efficiency of 92 percent for more than twelve months, has demonstrated the reliability of the aeroplane service. All the experiments prove that the universal use of the aeroplane for mail-carrying purposes is coming, and almost daily the papers chronicle the institution of new air lines. The collection of aerial post stamps thus becomes imperative to the up-to-date philatelist. One prophesies the not-far-distant date when the majority of the new stamps, certainly European, occupying the pages of our albums, till be those used in connection with the post conveyed by petrol-driven Mercuries. Tonight, time will only permit a short flight through the intensely fascinating history of the air mail, whilst I pilot you amongst the forty odd stamps which the past three and a half years have brought forth in this connection. The experience gained in the past War has, of course, been an invaluable help in the development of air services, and, curiously enough it was in wartime that the first airpost was instituted. It was necessary, during the Siege of Paris in 1870, to find a means of communication with the outer world, and, for this purpose, a balloon post was brought into being. The first ascent was made on 23rd September, 1870, and the services continued in almost daily use for four months, during which period 68 balloons were despatched, 60 landing on French or neutral territory, five being captured by the Germans, and three being lost at sea. Envelopes despatched in this manner give no indication of their mode of conveyance. However, it may be taken that any envelope or card bearing a postmark dated between 23rd September, 1870, and 28th January, 1871, was forwarded from Paris by balloon post - the first authoritative air mail. A connecting link between balloon and aeroplane posts took place in 1896, when a Mr. Fricker inaugurated a pigeon service between Great Barrier Island and Auckland N.Z., 66 miles apart, a post which continued for several years. Special triangular stamps were used for this service, depicting a pigeon in full flight, the denomination being 6d. (blue) and 1s. (red). The first aeroplane post in the world, however, took place on 18th February, 1911, at Allahabad, India, organised by Captain Windham. The letters were carried by aeroplane from the United Provinces Exhibition to a Post Office receiving-station in Allahabad, from which place they were despatched to any part of the world to which they were addressed. Over 6,000 letters and cards, thus posted, were franked by the Exhibition P.O. with a die, specially cut in the postal workshops at Aligarh, incorporating a design of an aeroplane, encircled by the inscription "Aerial Post, Allahabad Exhibition," together with the date of despatch. A nominal additional fee of six annas per letter or card was charged, which amount was handed, without deduction, as a donation to the new buildings of the Oxford and Cambridge Hostel at Allahabad. In honour of King George's Coronation this same Captain Windham was also the organiser of the first air post in the United Kingdom, which was flown between Hendon and Windsor on 9th September, 1911, and for a few following days. No special stamps were issued, but envelopes and postcards bearing a design of an aeroplane flying over Windsor Castle, with the winding Thames and St. Paul's Cathedral in the distance, were sold at 1s. and 6d. respectively. The postmark was worded "First United Kingdom Aerial Post," and the date; about 100,000 pieces of mail being carried by this service. U.S.A. was busy just about the same time, in 1911, experimenting with air mails, and this, coupled with the knowledge gained in the War, resulted in the establishment early in 1918, of an air line between New York and Washington, 218 miles apart. After the Armistice, lack of trains and engines led to an extension of this service to Cleveland and Chicago. This line now continues right on to San Francisco, by way of Omaha, Nebraska, Cheyenne, Wyoming, and Salt Lake City, Utah. The trip takes three days and is 2,651 miles in length, but results in letters reaching their destination 42 hours before the mail-train is due. Another air line runs between St. Louis, Chicago, and St. Paul, whilst numerous other towns are in process of being linked up by air. In the past twelve months over half a million pounds weight of mail matter has been airborne, and about £50,000 has been actually saved, as compared with the cost of transit by rail. On the 2nd of June, 1912, the Japanese postal authorities experimented with an air mail between Yokohama and Tokyo. A few letters are known to exist, whose envelopes bear the ordinary stamp and obliteration, with a special postmark, inscribed "Japanese Aerial Mail" and the Japanese equivalent for the date, but the attempt, being only experimental, was discontinued after the first day. From 1912 till 1917, aerial mails did not make much progress, but the reason which caused the inception of the air post, namely, war, was responsible for the re-opening of this means of communication. It happened that there was very serious congestion on the Italian railways in 1917, to relieve which an air mail was organised, on the 22nd May, between Rome and Turin. These cities are about 350 miles apart, the air space between them being bridged within four hours. In this connection. Italy achieved fame by being the first country to issue a stamp for use of its air mail, which took the form of an overprint on the 1903 "Inland Express Letter" stamp, 25c. rose, as follows :- ESPERIMENTO POSTA AEREA MAGGIO 1917 TORINO-ROMA - ROMA-TORINO

A month later, on 28th June, owing to the interference of Austrian submarines with Naples and Palermo, Sicily, mail steamers, a special seaplane service was inaugurated between these two places, which are 170 miles apart. The stamp used in this connection was the then unissued 40c. violet "Express Delivery" stamp, overprinted with the words IDROVOLANTE NAPOLl PALERMO NAPOLI 25 CENT 25

Another wartime air mail was brought into being on 30th March, 1918, by Austria, her planes carrying letters from Vienna to Kieff , with calls at Cracow and Lemberg. Three of the 1916 "Arms" type stamps were used, all being overprinted in block capitals with the formidable word "FLUGPOST", meaning "flying post."

"1.50 K 1.50'' was surcharged on the 2kr. (lilac) and "2.50 K 2.50" on the 3kr. (bistre), whilst the 4kr. (grey) was used without any surcharge. To the U.S.A. falls the honour of issuing the first distinctive air post stamp, which made its appearance on the 15th May, 1918, on the inauguration of the New York-Washington service. This stamp, which was recess-plate printed in carmine and blue, without watermark, at the Washington Bureau of Engraving and Printing, and is perforated 11, depicts a mail-carrying plane in flight. Over two million of these stamps were printed, out of which one lucky purchaser secured, over the P.O. counter, a sheet of 100 with the aeroplane inverted , the only sheet known to be printed in error. A Colonel Green eventually bought up the whole sheet, selling half of it, and retaining the remaining 50 in his own collection. These he had with him on hoard his steam yacht when it foundered in 1919, 43 of these stamps being so damaged as to be useless, which makes the error a very rare stamp. The air-mail fee was reduced on 15th July to 16 cents, and again in December, 1918 to 6 cents, for which stamps of these values were issued, yellow-green in the first instance, orange in the latter, the original design being retained. Since then the extra air-post fee has been abolished, and the ordinary mail fee of 2 cents, or 1d., an ounce is charged. It is not contemplated to issue a separate aerial mail stamp. On 4th July, 1918, the Hungarians experimented with an air mail, having Budapest as its starting point, and with various internal towns as destinations; but owing to the weather conditions and accidents to aviators, it was only in existence 20 days. Two of the 1916-17 stamps were surcharged as follows :- "1 K 50f. " on 75 filler (blue). "2 K 50f." on 2 krona (brown). the words "REPÜLÖ POSTA" being overprinted above the value, in red and blue respectively. This overprinting was carried out at the State Printing Works at Budapest, and is not remarkable for its good workmanship. This will be especially noticeable in the copy I show of the lower value, in which several letters are broken, whilst the "P" of "POSTA" has no top at all. It rather looks as if this stamp had taken part in one of the accidents which occurred. In October, 1920, an aerial mail was established, linking Hungary with other European countries, and the 1916-17 10kr. stamp was overprinted with the words "LEGI POSTA" and the new value, 3, 8, or 12 korona.

The air fairly hummed in 1919 with air post developments, and in all parts of the world aerial mails were instituted or carried out with this object in view. Early in the year an aeroplane post was started between Bombay and Karachi to expedite delivery of mail brought by steamers to Bombay. Public apathy and lack of support, however, caused the speedy discontinuance of this air line. Alexandria, Cairo, and Ismailia were linked together by air mail of 17th March, by "R.A.F." planes, which carried only official correspondence during the native disturbances, no special stamps or postmark being used. The envelopes, however, were marked with rubber stamp, "Aerial Post, E.E.F.", meaning "Egyptian Expeditionary Force". This service was discontinued as soon as conditions were normal. Switzerland was next in the field, or air, I should say, by the opening of a summer aerial post between Zurich and Lausanne, with calls at Berne and Neuchâtel, which took place on 28th April, 1919. The ordinary postage was charged, plus an air fee of 50c. For this latter purpose the current 50c. "Helvetia " type stamp was overprinted at the Federal Mint, Berne, with a design in red, showing the Swiss Military Air Force badge. Postmarks bearing the words "Schweiger Flugpost" were used, in addition to the ordinary express letter postmark. On the 5th May, 100,000 copies of the 35c. stamp of the 1906 issue of Tunis were ready for sale in connection with the air service which connected Gabés, Djerba, Zarzis, and Ben Gardane, as from that date. These stamps had been overprinted at the French Government Printing Works in Paris, and, in addition to the central overprint of the French aviator's badge, the air fee denomination of "30" centimes appeared on the stamp with the words "POSTE AÉRIENNE", the old value being obliterated by three bars. The stamp depicts the ruins of Hadrian's Aqueduct and, with its overprint, shows a true blending of the ancient with the modern. I wonder what Hadrian would say if he knew? This stamp has recently been replaced by a 30c. stamp of similar design, in blue and grey-green. We now come to the gallant, but unsuccessful attempt of Messrs. Hawker and Grieve to fly the Atlantic from Newfoundland to the United Kingdom. This took place on the 18th May, 1919, on a Sopwith machine, and resulted in the aeroplane falling into the sea, the two aviators with their mail being fortunately salved by a passing steamer. The mail reached the P.O. intact with the help of the British Fleet. The contents were undamaged, though in some cases wet, but none was in such condition as to prevent ultimate delivery. For the purposes of this mail 200 of the 3c. "Caribou" issue were overprinted at the Royal Gazette Office, St. John's (where all the other air-stamp overprints have been carried out), with the words "FIRST TRANSATLANTIC AIRPOST, April, 1919." - for the flight was expected to take place in April, though weather conditions were unfavourable until the following month. Of the 200 stamps, 18 were damaged and destroyed in the presence of the Auditor-General, 11 were used as presentation copies (one of which was sent to H.M. the King), and 95 were used and cancelled in the mail itself, leaving 76 still to be accounted for. These were sold at $25 each on behalf of the Marine Disaster Fund. and as only 182 of these stamps are known to exist, they are of great rarity. The first Trans-Atlantic Air Stamp was presented by the aviators, to be auctioned for the benefit of the Marine Disaster Fund. Lieut.-Col. E. S. Halford. of the Air Ministry, eventually bought the stamp for £210. Later in the month of May the air mail, ready for despatch by the Raynham-Martinsyde Atlantic flight, bore stamps of the 1c., 2c., 3c., and 24c. current "Caribou" series. These were overprinted as follows :-"1st Atlantic Airpost, Martinsyde-Raynham, Morgan". The cheers of the send-off had hardly died away before the plane crashed to earth to become a useless wreck, and the mail had to be despatched through the usual channels. The 15c. stamp of the 1897 (Jubilee) series, surcharged "Trans-Atlantic AIR POST, 1919. ONE DOLLAR" was now issued to prepay postage on letters sent by the Alcock-Whitten Brown flight to U.K. This non-stop flight commenced on 14th June, 1919, in a Vickers-Vimy machine, and by this means mail posted in Newfoundland on the early morning of 14th June was delivered in London on the night of the 17th-three days after leaving Newfoundland. The stamps were sold at $1 each, but the limited edition, was speedily bought up. 10,000 were surcharged in sheets of 25, making 400 sheets in all. In the overprinting errors hme crept in. Each sheet, therefore, contained 16 stamps normally overprinted; seven stamps with no comma after "POST"; one with an imperfect comma; and one without the full stop after "1919" and no comma after "POST". Thus it will be seen that of the 10,000 stamps issued, 6,400 were normally overprinted, 2,800 had no comma, 400 had an imperfect comma, and 400 had no stop or comma. It will be noticed that the block of four stamps, which I show, contains all four varieties - a rare combination. To celebrate an experimental air post between Puerto, Port Colombia, and Barranquila, 200 of the 1917 2c. Colombian stamps were overprinted locally with the inscription "1 en SERVICIO POSTAL AEREO 6-18-19" in five lines in black. Only one flight was made and the stamps were not accepted by the P.O. and were never cancelled by them. In October, 1920, an attempt was made to institute an air service between Cartagena and Barranquila, but owing to serious fatal accidents, this air mail has been indefinitely suspended. The contract for this service was given by the P.O. to a local firm, and letters carried through the air travelled at ordinary postage, plus 10c. per 15 grammes. Two thousand copies of a 10c oblong stamp were printed, depicting a vessel on the sea, with aeroplane above, and setting sun on the horizon. This was superseded by a set of seven values, issued privately by the air contractors, the design showing a map of the Colombian coast, with aeroplane in flight. To signalise the first air mail over the Rocky Mountains the envelopes of letters thus conveyed were franked with a special postmark bearing the words "1st B.C. Alberta Aerial Post." The mail in question was carried by plane, on August 5th, 1919, from Vancouver to Calgary, via Vernon and over the Great Divide to Lethbridge. On the return journey the pilot was forced to descend at Golden, and the letters were sent on by rail. Japan had made no serious attempt since its 1912 experiment to commence an air post, but with the intention of instituting regular flights between Tokyo and Osaka nearly 300 miles, stamps were issued for use on letters to be conveyed by the first air mail on October 3rd, 1919. These stamps were the current 1½ sen (blue) and the 3 sen (carmine), overprinted with the design of an aeroplane in red and black respectively. These two air stamps were on sale only at Head Post Offices on October 3rd, and in spite of elaborate precautions to prevent one person buying more than two stamps of each value, the entire issue of 40,000 overprinted stamps was sold out in a very short time. The weather played an important part in connection with this mail, and behaved so badly for day after day from October 4th, that the flight was abandoned for a further attempt (to the delight of the more superstitious Japanese and the letters sent by the usual method. The "King Albert Aerial Mail Service" was commenced early in January), 1920, in Belgian Congo. This service, which is carried out by seaplanes, embraces the whole of the Upper Congo River, and is flown in conjunction with the arrival of the Belgian mail steamers. In August last, four finely-drawn stamps, depicting scenes in the Congo with a seaplane above, made their appearance for use in this connection. The perforation is 12 and the values are :- 50c. orange and black. 1fr. violet and black. 2f blue and black. 5fr. green and black. By some unfortunate mistake, one which has caused the Belgian Government much annoyance, the word printed at the foot of each stamp, "Postluchtdienst," should have appeared as "Luchtpostdienst." As it stands the translation reads: "Service of the Postal Air," instead of "Postal Service of the Air." Of course, the printer may have been a man of imagination, and this was his way of prophesying that the air was soon to be so impregnated with correspondence as almost to describe the term "postal air." The air post instituted between Reval and Helsingfors in Estonia. was the direct outcome of the icebound nature of that country's coastline in the Gulf of Finland, which, at the time, permitted only a few Ships to arrive at Reval. Thus it happened that, on 7th February, 1920, three British-piloted planes left Reval with mail and reached Helsingfors in less than an hour later. Weather conditions prevented the return journey being made for over a week. The service, however, was continued until two months later, when, owing to a shortage of aeroplanes, only a very small proportion of the mail could be carried. Preference was given to diplomatic and Registered letters, ordinary being taken if there was room. The breaking up of the ice early in May permitted the re-opening of sea communication, and the air mail was discontinued. In March a five mark imperforate triangular stamp was issued for use on this mail, printed in yellow, blue and black, and showing an aeroplane in flight. The ordinary postage was charged in addition to the air fee. A Tientsin-Pekin aerial mail was inaugurated as a regular service on 7th May, 1920, with Handley-Page machines. Letters posted at 5 p.m. in Tientsin can now be delivered in Pekin three hours later. No special stamp has so far been issued, but the postmark reads: - "Chinese Post Office - despatched by aeroplane - Tientsin-Pekin." The Chinese Cabinet has now sanctioned the opening of an air service between Pekin and Shanghai, with three intermediate stations, and 80 landing grounds. Siam, a country whose airmen are so intrepid and so seemingly without nerves, has commenced in September an aerial post between Bangkok and Chantzboon, roughly 300 miles. The current 5 satangs stamps has been overprinted by hand-machine, with the Siamese emblematic bird, the garuda, under which appear four lines of native wording. When the London-Paris Airpost was opened to the general public on 10th November, 1919, the charge was excessive, viz, 2s. 6d. per ounce. The total number of letters sent on the first day after this charge was made totalled 315, whilst the aeroplanes ready for use had a capacity of 76,000 letters. Since then, however, steps have been taken to popularise the aerial mail, the chief of which has been the reduction of the air fee to 2d. per ounce, plus ordinary postage, whilst the express fee is 6d. an ounce. A blue label, inscribed "BY AIR MAIL," which can be obtained free from the Post Office, is the only outward and visible token on the left-hand corner of the envelope that it has travelled from England by aerial mail. The absence of this label, so long as the envelope is clearly marked as to its means of conveyance, will not debar the letter from being forwarded by air and delivered. In France, a label is attached to the envelope, depicting as its central design, the great French aviator, Guynemer. Besides the twice daily service to Paris and back, carried out by the Aircraft Transport and Travel Co., Ltd., recent air lines have linked together six countries, namely, England, Holland, Germany, Denmark, Sweden, and Norway. This service was inaugurated on 15th September, when a Danish-piloted de. Haviland plane left Copenhagen with the London mail and, travelling via Hamburg, reached Amsterdam, where the letters were transferred to the Handley-Page and Airco joint air service, and so to London. On the return journey, Queen Alexandra sent a basket of fruit to the Empress Dowager of Russia. The fruit left London a 3p.m. and was conveved to Her Majesty, outside Copenhagen, by 5 p.m. the following day. Last week a mail plane flew from Cricklewood, London, made a stop at Amsterdam, and arrived at Copenhagen in 5 hours 40 minutes, the distance being 520 miles. The London-Amsterdam service, instituted on 5th July, 1920, is carried out by the Handley-Page Transport Co., and the journey of 265 miles has been flown in 1 hour 50 minutes, or an average of 150 miles per hour. For use in connection with this inking up of countries, Sweden, at the end of September, issued three overprinted stamps, viz. :- 3 öre brown, Official, surcharged "LUFTPOST 10" 2 öre orange, Official, surcharged "LUFTPOST 20" 4 öre lilac, Official, surcharged "LUFTPOST 50" Envelopes bear a blue label, similar to that of Great Britain, but the word "LUFTPOST" is printed in red thereon. A provisional overprinting of the 1, 5, 10, 25, and 50c. current Spanish stamps with the words "CORREO AEREO," marked the opening of an aerial post between Seville and Larache, in Morocco; between Barcelona and Palma, Morocco; and between Malaga and Melilla, on April 4th. Only 20,000 sets were issued and these provisionals are to be superseded at an early date, by a distinctive series of air stamps, portraying the progress of aerial navigation. One of the latest countries to send mail through the air is the go-ahead State of Czecho-Slovakia. Three of the Hradschin series of stamps have been surcharged with new values, whilst a design of an aeroplane now forms the centre of the stamp. The 200 heller value is surcharged "14 KRONES," which is the ordinary postage plus air fee between Prague and Warsaw. The 500 heller bears a new value of 24 krones, for use between Prague and Paris, a 5½ hours' journey, carried out thrice weekly, travelling via Strashourg. The planes are sufficiently roomy to allow the carrying of passengers and goods. The 1,000 heller now takes the value of 28 krones, for use between Prague and London. On October 16th Danzig advertised its air mail by the issue of three provisionals. These stamps were the 40pf. Germans, already overprinted "DANZIG"; and further overprinted with new denominations, 40 and 60pf. and 1 mark, together with the design of an aeroplane on the two lower values and a winged posthorn on the 1 mark. From this brief survey, of the development of the aerial mail, it will be admitted, I think, that the prophecy contained in my opening remarks as to the coming of universal air posts, is well-founded - or well-aired, whichever is the correct term. Only a week or two ago the newspapers reported a combination of seven air transport firms, British, Danish, Dutch, Swedish, French, Roumanian, and German, with a view to completing a network of air lines that will shortly spread over the whole of North-West Europe. A new world to conquer has sprung up before the philatelist, one in which his imagination, initiative and foresight can play an important part, and I trust that my remarks this evening may prove of use to those whose flight of fancy take them into the ethereal realms of aerial philately. Read the full article

[europeconsalting-blog]

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[postsofbabel]

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[christusleeft]

New Post has been published on In de hemel is wél bier !

New Post has been published on http://bit.ly/2lLtZPe

1Petrus 1

Opschrift en zegengroet

1 Petrus, een apostel van Jezus Christus, aan de vreemdelingen, averstrooid in Pontus, Galatie, Kappadocie, Azie en Bithynie,

a: Jas 1:1  Jakobus, een dienstknecht van God en van den Heere Jezus Christus; aan de twaalf stammen, die in de verstrooiing zijn: zaligheid. 

2 Den uitverkorenen naar de voorkennis van God den Vader, in de heiligmaking des Geestes, tot gehoorzaamheid en bbesprenging des bloeds van Jezus Christus; cgenade en vrede zij u dvermenigvuldigd.

b: Heb 12:24  En tot den Middelaar des nieuwen testaments, Jezus, en het bloed der besprenging, dat betere dingen spreekt dan Abel. 

c: Rom 1:7  Allen, die te Rome zijt, geliefden Gods, en geroepen heiligen, genade zij u, en vrede van God, onzen Vader, en den Heere Jezus Christus. 1Co 1:3  Genade zij u en vrede van God onzen Vader, en den Heere Jezus Christus. Gal 1:3  Genade zij u en vrede van God den Vader, en onzen Heere Jezus Christus; Eph 1:2  Genade zij u en vrede van God, onzen Vader, en den Heere Jezus Christus. 

d: 2Pe 1:2  Genade en vrede zij u vermenigvuldigd door de kennis van God, en van Jezus, onzen Heere; Jud 1:2  Barmhartigheid, en vrede, en liefde zij u vermenigvuldigd. 

De hoop der zaligheid

3 eGeloofd zij de God en Vader van onzen Heere Jezus Christus, fDie naar Zijn grote barmhartigheid ons heeft wedergeboren, tot een levende hoop, gdoor de opstanding van Jezus Christus uit de doden.

e: 2Co 1:3  Geloofd zij de God en Vader van onzen Heere Jezus Christus, de Vader der barmhartigheden, en de God aller vertroosting; Eph 1:3  Gezegend zij de God en Vader van onzen Heere Jezus Christus, Die ons gezegend heeft met alle geestelijke zegening in den hemel in Christus. 

f: Rom 6:23  Want de bezoldiging der zonde is de dood, maar de genadegift Gods is het eeuwige leven, door Jezus Christus, onzen Heere. Jas 1:18  Naar Zijn wil heeft Hij ons gebaard door het Woord der waarheid, opdat wij zouden zijn als eerstelingen Zijner schepselen.

g: 1Co 15:20  Maar nu, Christus is opgewekt uit de doden, en is de Eersteling geworden dergenen, die ontslapen zijn. 

4 Tot een onverderfelijke, en onbevlekkelijke, en onverwelkelijke erfenis, hdie in de hemelen bewaard is voor u,

h: Col 1:5  Om de hoop, die u weggelegd is in de hemelen, van welke gij te voren gehoord hebt, door het Woord der waarheid, namelijk des Evangelies; 2Ti 1:2  Aan Timotheus, mijn geliefden zoon: genade, barmhartigheid, vrede zij u van God den Vader, en Christus Jezus, onzen Heere.

5 Die in de kracht Gods bewaard wordt door het geloof tot de zaligheid, die bereid is, om geopenbaard te worden in den laatsten tijd. 6 iIn welke gij u verheugt, knu een weinig [tijds] (zo het nodig is) bedroefd zijnde door menigerlei verzoekingen;

i: Rom 5:3  En niet alleenlijk dit, maar wij roemen ook in de verdrukkingen, wetende, dat de verdrukking lijdzaamheid werkt; Jas 1:2  Acht het voor grote vreugde, mijn broeders, wanneer gij in velerlei verzoekingen valt; 

k: Heb 10:37  Want: Nog een zeer weinig tijds en Hij, Die te komen staat, zal komen, en niet vertoeven. 1Pe 5:10  De God nu aller genade, Die ons geroepen heeft tot Zijn eeuwige heerlijkheid in Christus Jezus, nadat wij een weinig tijds zullen geleden hebben, Dezelve volmake, bevestige, versterke, en fondere ulieden. 

7 lOpdat de beproeving uws geloofs, die veel kostelijker is dan van het goud, hetwelk vergaat en door het vuur beproefd wordt, bevonden worde te zijn tot lof, en eer, en heerlijkheid, in de openbaring van Jezus Christus;

l: Isa 48:10  Ziet, Ik heb u gelouterd, doch niet als zilver, Ik heb u gekeurd in den smeltkroes der ellende. 1Co 3:13  Eens iegelijks werk zal openbaar worden; want de dag zal het verklaren, dewijl het door vuur ontdekt wordt; en hoedanig eens iegelijks werk is, zal het vuur beproeven. Jas 1:3  Wetende, dat de beproeving uws geloofs lijdzaamheid werkt. 1Pe 4:12  Geliefden, houdt u niet vreemd over de hitte der verdrukking onder u, die u geschiedt tot verzoeking, alsof u iets vreemds overkwame;

8 mDenwelken gij niet gezien hebt, en [nochtans] liefhebt, in Denwelken gij nu, hoewel [Hem] niet ziende, maar gelovende, u verheugt met een onuitsprekelijke en heerlijke vreugde;

m: Joh 20:29  Jezus zeide tot hem: Omdat gij Mij gezien hebt, Thomas, zo hebt gij geloofd; zalig zijn zij, die niet zullen gezien hebben, en nochtans zullen geloofd hebben. 

9 Verkrijgende het einde uws geloofs, [namelijk] de zaligheid der zielen. 10 Van welke zaligheid ondervraagd en onderzocht hebben nde profeten, die geprofeteerd hebben van de genade, aan u [geschied];

n: Gen 49:10  De schepter zal van Juda niet wijken, noch de wetgever van tussen zijn voeten, totdat Silo komt, en Denzelven zullen de volken gehoorzaam zijn. Dan 2:44  Doch in de dagen van die koningen zal de God des hemels een Koninkrijk verwekken, dat in der eeuwigheid niet zal verstoord worden; en dat Koninkrijk zal aan geen ander volk overgelaten worden; het zal al die koninkrijken vermalen, en te niet doen, maar zelf zal het in alle eeuwigheid bestaan. Hag 2:8  Ja, Ik zal al de heidenen doen beven, en zij zullen komen tot den Wens aller heidenen, en Ik zal dit huis met heerlijkheid vervullen, zegt de HEERE der heirscharen. Zec 6:12  En spreek tot hem, zeggende: Alzo spreekt de HEERE der heirscharen, zeggende: Ziet, een Man, Wiens naam is SPRUITE, Die zal uit Zijn plaats spruiten, en Hij zal des HEEREN tempel bouwen. 

11 oOnderzoekende, op welke of hoedanigen tijd de Geest van Christus, Die in hen was, beduidde en te voren getuigde, phet lijden, [dat] op Christus [komen zou], en de heerlijkheid daarna [volgende].

o: Dan 9:24  Zeventig weken zijn bestemd over uw volk, en over uw heilige stad, om de overtreding te sluiten, en om de zonden te verzegelen, en om de ongerechtigheid te verzoenen, en om een eeuwige gerechtigheid aan te brengen, en om het gezicht, en den profeet te verzegelen, en om de heiligheid der heiligheden te zalven.

p: Psa 22:7  Maar ik ben een worm en geen man, een smaad van mensen, en veracht van het volk. Isa 53:3  Hij was veracht, en de onwaardigste onder de mensen, een Man van smarten, en verzocht in krankheid; en een iegelijk was als verbergende het aangezicht voor Hem; Hij was veracht, en wij hebben Hem niet geacht. 

12 Denwelken geopenbaard is, dat zij niet zichzelven, maar ons bedienden deze dingen, die u nu aangediend zijn bij degenen, die u het Evangelie verkondigd hebben qdoor den Heilige Geest, Die van den hemel gezonden is; rin welke dingen de engelen begerig zijn in te zien.

q: Act 2:4  En zij werden allen vervuld met den Heiligen Geest, en begonnen te spreken met andere talen, zoals de Geest hun gaf uit te spreken. 

r: Eph 3:10  Opdat nu, door de Gemeente, bekend gemaakt worde aan de overheden en de machten in den hemel de veelvuldige wijsheid Gods; 

Opwekking tot een heiligen wandel

13 sDaarom opschortende de lenden uws verstands, [en] nuchteren zijnde, hoopt volkomenlijk op de genade, die u toegebracht wordt in de openbaring van Jezus Christus.

s: Luk 12:35  Laat uw lendenen omgord zijn, en de kaarsen brandende. Eph 6:14  Staat dan, uw lenden omgord hebbende met de waarheid, en aangedaan hebbende het borstwapen der gerechtigheid; 

14 Als gehoorzame kinderen, wordt niet gelijkvormig aan de begeerlijkheden, die te voren in uw onwetendheid waren; 15 Maar gelijk Hij, Die u geroepen heeft, heilig is, t[zo] wordt ook gijzelven heilig in al [uw] wandel;

t: Luk 1:75  In heiligheid en gerechtigheid voor Hem, al de dagen onzes levens. 

16 Daarom dat er geschreven is: vZijt heilig, want Ik ben heilig.

v: Lev 11:44  Want Ik ben de HEERE, uw God; daarom zult gij u heiligen, en heilig zijn, dewijl Ik heilig ben; en gij zult uw ziel niet verontreinigen aan enig kruipend gedierte, dat zich op de aarde roert. Lev 11:45  Want Ik ben de HEERE, die u uit Egypteland doe optrekken, opdat Ik u tot een God zij, en opdat gij heilig zijt, dewijl Ik heilig ben. Lev 19:2  Spreek tot de ganse vergadering der kinderen Israels, en zeg tot hen: Gij zult heilig zijn, want Ik, de HEERE, uw God, ben heilig! Lev 20:7  Daarom heiligt u, en weest heilig; want Ik ben de HEERE, uw God! 

17 En indien gij tot een Vader aanroept Dengene, xDie zonder aanneming des persoons oordeelt naar eens iegelijks werk, zo wandelt in vreze den tijd uwer inwoning;

x: Deu 10:17  Want de HEERE, uw God, is een God der goden, en een Heere der heren; die grote, die machtige, en die vreselijke God, Die geen aangezicht aanneemt, noch geschenk ontvangt; 2Kr 19:7  Nu dan, de verschrikking des HEEREN zij op ulieden; neemt waar, en doet het; want bij den HEERE, onzen God, is geen onrecht, noch aanneming van personen, noch ontvanging van geschenken. Job 34:19  Hoe dan tot Dien, Die het aangezicht der vorsten niet aanneemt, en den rijke voor den arme niet kent? Want zij zijn allen Zijner handen werk. Act 10:34  En Petrus, den mond opendoende, zeide: Ik verneem in der waarheid, dat God geen aannemer des persoons is; Rom 2:11  Want er is geen aanneming des persoons bij God. Gal 2:6  En van degenen, die geacht waren, wat te zijn, hoedanigen zij eertijds waren, verschilt mij niet; God neemt den persoon des mensen niet aan; want die geacht waren, hebben mij niets toegebracht. Eph 6:9  En gij heren, doet hetzelfde bij hen, nalatende de dreiging; als die weet, dat ook uw eigen Heere in de hemelen is, en dat geen aanneming des persoons bij Hem is. Col 3:25  Maar die onrecht doet, die zal het onrecht dragen, dat hij gedaan heeft; en er is geen aanneming des persoons.

18 yWetende dat gij niet door vergankelijke dingen, zilver of goud, verlost zijt uit uw ijdele wandeling, die [u] van de vaderen overgeleverd [is];

y: 1Co 6:20  Want gij zijt duur gekocht: zo verheerlijkt dan God in uw lichaam en in uw geest, welke Godes zijn. 1Co 7:23  Gij zijt duur gekocht, wordt geen dienstknechten der mensen. 

19 Maar zdoor het dierbaar bloed van Christus, als van een onbestraffelijk en onbevlekt Lam;

z: Act 20:28  Zo hebt dan acht op uzelven, en op de gehele kudde, over dewelke u de Heilige Geest tot opzieners gesteld heeft, om de Gemeente Gods te weiden, welke Hij verkregen heeft door Zijn eigen bloed. Heb 9:12  Noch door het bloed der bokken en kalveren, maar door Zijn eigen bloed, eenmaal ingegaan in het heiligdom, een eeuwige verlossing teweeggebracht hebbende. Rev 1:5  En van Jezus Christus, Die de getrouwe Getuige is, de Eerstgeborene uit de doden, en de Overste der koningen der aarde. Hem, Die ons heeft liefgehad, en ons van onze zonden gewassen heeft in Zijn bloed. 

20 Dewelke wel avoorgekend is geweest voor de grondlegging der wereld, maar geopenbaard is in deze laatste tijden om uwentwil,

a: Rom 16:25  Hem nu, Die machtig is u te bevestigen, naar mijn Evangelie en de prediking van Jezus Christus, naar de openbaring der verborgenheid, die van de tijden der eeuwen verzwegen is geweest; Eph 1:9  Ons bekend gemaakt hebbende de verborgenheid van Zijn wil, naar Zijn welbehagen, hetwelk Hij voorgenomen had in Zichzelven. Eph 3:9  En allen te verlichten, dat zij mogen verstaan, welke de gemeenschap der verborgenheid zij, die van alle eeuwen verborgen is geweest in God, Welke alle dingen geschapen heeft door Jezus Christus; Col 1:26  Namelijk de verborgenheid, die verborgen is geweest van alle eeuwen en van alle geslachten, maar nu geopenbaard is aan Zijn heiligen; 2Ti 1:9  Die ons heeft zalig gemaakt, en geroepen met een heilige roeping; niet naar onze werken, maar naar Zijn eigen voornemen en genade, die ons gegeven is in Christus Jezus, voor de tijden der eeuwen; Tit 1:2  In de hoop des eeuwigen levens, welke God, Die niet liegen kan, beloofd heeft, voor de tijden der eeuwen, maar geopenbaard heeft te Zijner tijd; 

21 Die door Hem gelooft in God, Welke Hem opgewekt heeft uit de doden, ben Hem heerlijkheid gegeven heeft, opdat uw geloof en hoop op God zijn zou.

b: Act 2:33  Hij dan, door de rechter hand Gods verhoogd zijnde, en de belofte des Heiligen Geestes, ontvangen hebbende van den Vader, heeft dit uitgestort, dat gij nu ziet en hoort. Php 2:9  Daarom heeft Hem ook God uitermate verhoogd, en heeft Hem een Naam gegeven, welke boven allen naam is; 

22 Hebbende [dan] uw zielen gereinigd in de gehoorzaamheid der waarheid, door den Geest, ctot ongeveinsde broederlijke liefde, zo hebt elkander vuriglijk lief uit een rein hart;

c: Rom 12:10  Hebt elkander hartelijk lief met broederlijke liefde; met eer de een den ander voorgaande. Eph 4:3  U benaarstigende te behouden de enigheid des Geestes door den band des vredes. Heb 13:1  Dat de broederlijke liefde blijve. 1Pe 2:17  Eert een iegelijk; hebt de broederschap lief; vreest God; eert den koning. 

23 Gij, die dwedergeboren zijt, niet uit vergankelijk, maar [uit] onvergankelijk ezaad, door het levende en eeuwig blijvende Woord van God.

d: Jas 1:18  Naar Zijn wil heeft Hij ons gebaard door het Woord der waarheid, opdat wij zouden zijn als eerstelingen Zijner schepselen. 

e: 1Jn 3:9  Een iegelijk, die uit God geboren is, die doet de zonde niet, want Zijn zaad blijft in hem; en hij kan niet zondigen, want hij is uit God geboren.

24 fWant alle vlees is als gras, en alle heerlijkheid des mensen is als een bloem van het gras. Het gras is verdord, en zijn bloem is afgevallen;

f: Isa 40:6  Een stem zegt: Roept! En hij zegt: Wat zal ik roepen? Alle vlees is gras, en al zijn goedertierenheid als een bloem des velds. 1Co 7:31  En die deze wereld gebruiken, als niet misbruikende; want de gedaante dezer wereld gaat voorbij. Jas 1:10  En de rijke in zijn vernedering; want hij zal als een bloem van het gras voorbijgaan. Jas 4:14  Gij, die niet weet, wat morgen geschieden zal, want hoedanig is uw leven? Want het is een damp, die voor een weinig tijds gezien wordt, en daarna verdwijnt. 1Jn 2:17  En de wereld gaat voorbij, en haar begeerlijkheid; maar die den wil van God doet, blijft in der eeuwigheid. 

25 Maar het Woord des Heeren blijft in der eeuwigheid; en dit is het Woord, dat onder u verkondigd is.

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LEE KNOW & CHANGBIN / 220627

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HAN & FELIX / 220704

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SEUNGMIN & HYUNJIN / 220418