TRU Mathematics 16’1 (1980〕
ON FERMAIT t S LAST THEOREト1 AND THE
FIRST FA㎝)R OF THE CLASS NUMBER OF
THE CYCI、C叩OMIC FIELD兀 Yoshikazu KARAMATSU (Received May 30, 1980〕 1. IntroduCtion. The author proved the fbllowing theorem [2]. 、HE。剛.写みyZ.み。i.。。ti。fi。砺』。。。 。.・,; z,、.i・M。,。協θ oca pime Z。仇θη疏θ方rsカfactor o了theθZα合S numわθr oアtheσycZotomie fieZd た(・)〔ζ・・2T「i/Z〕i・砺・ibZ・ by ZIU ・x・・輌h・・Z・3,・47,・・4,・・3,・・1. In this paper I intend to develop this result and prove that the fbllowing 〔a),〔b〕hold厄th some excepti㎝s:;蕊爵:,:㍗:,罐1誌㌶・韮品、:1霊};1°f Z2・
2.Ku皿ner Criteria and the Bernouユ1i Nun』ber. Prof. T. Wbrishima [4] showed that if Z is an odd prime and〔1) みηz・みo 、 ’』
is sati・fied桓reati鋤1−integ号rS pr㎞e t・ea(in・ ・ther ・nd t・Z・. then fOrn=2,3,…,(Z−1)/2 . 、 、
(・) @〔・2’・ 一・)b2n[裏.〔;:鋼・(・ )・.ゴ・・( 、)];・(・…2〕where
..
E,(_{竪]。。。, ..
−t・b・ing・・ny・f the ratig・ ’f3) 芸・÷・チる÷・÷
・・db。…ゐ、・一・/2,…,力劔一(一・)n−IBn。 b2n.、・・ar・血・・励・…fBemoulli. . 、 . ..
23
24
Y.KARAMATSU
For η=2,3,4,5,6,7,8,9,10 and Z>174611 we have (・en−・〕b、。・f・ (・・dZ〕・ Hence, the relation (2) gi廿es ・(、.2“)、.、・・、(。、.、)・・ 伽・・………〕〔・・d・2) that is〔・〕 ・,_、[d(当號、1元・㌢〕L.。・・(・・d・2)
(η∼=233..・.310; Z>174611). ・ti・㎞・wn・h…(、.。、)、.、・・(md・2)f・・・・・・・・…〔・・匝・・㎜・f・[・D and for m=6,7 〔T. Mbrishima [4] and D. H. Lehmer [3]) and s i皿ilarly fbr m=8 (Y. Karamatsu [2]〕. In the present paper, using 〔4〕,we shall extend these results. we have, for m=2,3,… ,8, 〔・〕@ [d(2m−’舞ll評]v.。・[ヂ;i繋fθ判炉。≠・
and for涜=9,10 ・(・〕 [’(等號、};・%L・[ヂ器…f判_・
〔mod Z〕,Where
田㌶㌶::莞:;;;蕊潔:ll:1蒜欝8・549t5
・6,382,798,925,47Stg・3,207,483,178,IS7tl°・782,115,518,299tll .85,383,238,549t・2・3,572,・85,255t13・4・,932,745み65,5・9t’5・t’6 (mod Z), 一P2n_ユ〔t) 2mLl 〔1−t〕蹴㌶i畿;ll:;;:㌶:1鵠1鵬;㌶::!;91g°°t5
・・,865,385,657,780,650tg・1,865,385,657,780,650t’°・1,006,709,967,915,228t” .285,997,・743・7,…t・2・4・,457,344,748,・72t’3・2,575,・22,・97,6・・t14 .6・,4・3,3・3,…t・5・382,439,924t16 ・ 262,・2Sti7・t⊥d. 舟w1・t u・c…ider th・p・1)m・㎡・1 P。ω・f d・gree・−1砿th n・tura1・励e「 coefficieIlts, forヵ=17,19. pn(x)i・divi・ib1・by苫(1−¢)・hence・鴨put Pη〔x)=虚(1−x)Qn〔苫〕・ON FERMA¥T , S LAST THEOREM 25 th・n・fb・x(1−・〕≠0〔・・d Z)・P。ω‘i・divi・ib1・by Z・nly wh・n Qn〔司i・ divisible by Z. 『 On the other hand, from the assulqption @,y,a,Z〕=1, we have, l z z =0 〔mod Z), x+y+9≡x +y 十9 therefbre, the s’iX values module Z of (3) aτe expTessed by the fOllowi皿g 一ち÷.一・・ち,i、,≒t,、…ピ The values
(・〕 ち÷,・一ち、≒,≒1,,喜、
are roots of the congruence 〔・〕 ・、(t)・t6−3・5・・…L−(・α一5)・3+αt2−3t・… 〔・・d・〕, wh。。e α.・一(・・3t・一 st3・・t5・t6〕. t2(1..t〕2 H・nce・qn〔t〕・an b・w・itt・n・・ (9) On〔t〕=o壬(t)9、〔t〕+R、〔t〕 wh・・e ai〔t)i・th・q・・ti・nt・ It is convenient to distinguish three cases as fbllows: Case A. The six values in(7)are relatively different modulo Z. Case B. The three values i皿 〔7〕 are co亘gruent to each other modulo Z. Case C. The two values in 〔7〕 are congruent to each other modUlo Z. First of a11, clearly t≠0,・1 〔mod Z〕. 」Case A: In this case we have t≠−1 (mod l). If%(t〕i・divi・ib1・by Z・th・n Rユ〔t〕皿・t b・diVi・ib1・by Z・ R、ぽ,〔・〕t5・へ〔・)t4・A,〔・)t3・A,〔・)t2・A、〔・〕・’・A。〔・〕, ’』・Ai(・〕(i・・,・,2,…,5), i・ap・・yn・mi…f・n・tur…umi)・r c・effi・i・n・・. Hence, for all i 傷〔・〕≡0 〔i=0・1・2・…・5) 〔m・d Z)・ From these six congruences, we have three congruences ㌧α+炉0〔i=1・2・3) 〔m・dZ)・ nh…hi ・nd ki ar・ s・m・ int・g・…26
Y.KAUIAMATSU
U・ing・these r・1・tiq鵬“⑭e⑭n屹rsカユ・、 b2・suCh『伽t
孤dゐ2・0(ftpd・Z〕・肋・e・ver・鳩see d違t th・greatest・・㎜n輌i・・r・・f
th・int・9r・・カ、・・d b2 f・・17㎝d 19 are・・Spectivelγユ7〔Y・、 (H.・Wada [8D. The 6quation 〔1) is not satisfied fbr Z=17 th・・efbre・w・ have P。(t)=t〔1一鰍〔カ〕≠0(mbd Z)fb・ Case B: ’Pollaczek 1[5] showed that this Case B never hapPens on both φn(t)Bl_7τ≡0 (lnod Z〕, π=3,5,・◆●,Z−2 and φZ_1〔t)三〇 (血od Z),wheτe z_1φn(t)=Σ〔−1〕m−1mn−ltm .・’
ηFl and so it is(luite unnecessary to consider this case. Hbwever, we can show th・t Qn〔t)fO(modZ)fb・・=17・19・・fb11・幡・linqe・the c・n・liti・n・・f Case. B, the values of〔7)are roots of the congruence〔・・) 9、(t)・t2・t・… 〔・・dZ)
砲ere tfO,1,−1,(modZ〕.
we have Qn〔t〕=%(t)92〔t)+Rfi〔カ)・ ・Where・Q;(t)i・th・q・・ti・㎡・. lf Qn〔t〕.i・di・i・ib1・by Z・then・RS(t)「「「ust be divisible by Z, alld we can、 see that 』 , Ri7=−5・515・342・166・891=−23’401’13・687’43・691≡0〔m°d Z〕fb「π=17・ Ri9=1・538・993・024・478・301=7’19’7691”8609’174・763≡0〔m°dt〕f°「 ・n=19・ 血e ・quati・n(1〕i…t・ati・fi・d fb・Z砲i・h』a・e th・p・㎞・.fa・t・r・。f Ri了 , since undet the asξumption of the Case I,〔1) is not satisfied fbrand R’ lg Z<253,749,889 〔D.H.1、ehmer [3D.・廠efb・・eh〔t)≠0〔・・17・19〕(・・d Z)・
Case C・・f…ド2,÷〔・・d・),・h㎝(・)・・{・,−2,一÷}・H・n・eQn〔’1)三〇〔mdZ){b・・=17・19飢d
Qn〔−2〕E −Qn(÷)(md Z〕・ Mbreover, we can get: Qユ7〔−2)=5・17’20・845・704・635・211≡0 〔m・dZ) . a、g〔−2〕=7・19・916・933’9・253・728・769三〇 (m°dZ〕 Therefbre we have fbr m=9, 10ドー11器三θ%L。≠・ (・…)
except when Z=20,845,704,635,211; Z=9,253,728,769. bl・.三〇〔・。dZ〕 Karamatsu) and 19 and Z=19. n=17, 19.ON FEiRMItT , S LAST THEORHM 27 From above results A, B and C we have the follow血g Le㎜a; LEWA 1・ヱアZisαn・・ld pri〃θαncl〔1)is sαtisfied in・αse 1, then forθα・h n∼=2,3,4,5,6}7,8,9 an〈牙 10 〔11〕 @ D(2η・−1)Z− [ d(肋;…1嘉;ユ 1−eVt〕 ] ∂=010 〔mod Z〕 exeePt tuhen Z α㌘θ the ゴ:b ZZoτ励η≦7 thTee pnimes 3,547,114,323,481; 20,845,704,635,221; 9,253,728,769, ∼Dhe?e−t heingαny oアthe eeαtios in (3〕. Now applying this Lemma to(4〕,we easily obtain the following theor㎝.
mEO剛1・写Zis an・dd P”ime and(1〕is sαti$fied in伽・1.カ』f・P eαeh
m=2,3,4,5,6,7,8,9,10〔・2) b(Z.n)、.、・・ (・・d Z2)
hoZds forαZZ Z3 hut pespeetivily for m=8,9,10 exoeρt Z whieh ave three エ ・・im・・ment・・n・d・i・・L・tnm…wh・・e・ユー一・/…、i−(一・)¢+’Bi. b、i.、一・. t・・, B. α㌘θ Bθヱη2(フulZi nz〃ibθ?8. 3.The first factor of the class number of the cyclotomic field. By H. S. Vandiner’s result [6]we have (・3)@今1妻已顯咽〔妻ザZ8一ユ
where
d。三一α/z 〔・・d・) 0≦dα<n 〔nJZ〕=1・ H・nce fd・i−(1−2m) zc・1,・・0, 〔・4〕@t−・ g¥1・’・橿巨αα垣 (m・d・・〕
On the other hand, it is㎞om that〔・5) ÷封。・・b、 (m。d・・)
α=1 where z<3.28
Y.KARAMATSU
Hence
〔16)For c=1
≒・b、・Si’ ・。・i−・ α=1 and 19, this yields n(Z_2m)z+・.1 わ 〔Z−2m)Z+1 (z−en)Zl9+1n
(mod Z2). (17〕 (Z−2m〕119・1ぽ、.、・
ワ・。・・(z−2m)z_1
カ(z.en)z・9+、 (mod Z2) 、S1、.d(・−an)z19 a α;1 〔mod Z2). Hence we have 。(z−・・)z。α(z−・・)z19 〔。。dZ2).Accordingly
〔・8)・
=G≒・・(_)、.、・・i惑;i}1・(・.・・)・・9+・ (・・d Z2〕・ For each η∼=2,3,4,5,6,7,8,9,10, we have 。(・一・・)・19+・担 ’(。。d、〕, ・ 油ence, fl℃皿 (18) and 〔12),we have〔・9〕 b(Z−、m)、・.ユ・・ 圃・2)
伽=2,3,…,10). F・㎝H.S.迦div・…rg・ult[7]・・nceming th・fi・・t fact・・hl・f th・ 2TTt class nuniber of k〔ζ), ζ=e τ ,we also have〔・・〕 h、・Z碁き…19+1 ⑭’g)
血ere μ=〔Z−1〕/2 and s=1,3,・.・,Z−2. Therefbre we obtain fr㎝ (19) and (20〕, h、・・ 〔・・dZ19〕・ Hence we have the following theorem.皿・剛2.ffみみみO i。。。ti。fied in vatiouaZ integ…x,y,・輌・
t・・ca・P・・ilm・Z. th・n th・ fi?・t fa・t・㌘・μん・・ZαSS nw・b・r・f仇・・y・Z・古・屹 fi。Zd・k〔ζ)。ζ・。2πi/Z. i。 di・i・ibZ・ by zl9・騨鋤・n Z…伽θ輌・・﹂ ()NFERMttT , S LAST THEOREM