A TABLE OF ABSOLUTE NORMS OF HEILBRONN SUMS

By Ken Yamamura

Department of Mathematics, National Defense Academy

Let p be an odd prime number and ζ a primitive p^{2}th root of unity. Let L be the unique subfield of Q(ζ) of degree p. The pth Heilbronn sum is defined as the trace of ζ from Q(ζ) to L. We denote by NH_{p} its absolute norm. Fouché[1] proved that if l is a prime divisor of NH_{p}, then l satisfies the congruence

l^{p−1} ≡ 1 (mod p^{2}).

This congruence is well known for l = 2, because Wieferich[4] proved that if there exists a counterexample to the first case of Fermat’s last theorem for the exponent p (FLT,I,p), then l = 2 satisfies the congruence above. Wieferich’s result has been generalized as follows [3]: if there exists a counterexample to (FLT,I,p), then all prime numbers l with 2 ≦ l ≦ 113 satisfy the congruence above. In [1], the table of the values of NH_{p} for p < 50 was given (by the referee). We extend it to p < 200 and give complete factorizations of NH_{p} except for p = 197 and p = 199. (‘c’ means composite.)

Remark. In [2] Ihara tentatively defines the differential dα of nonzero number α which is not a root of unity in an algebraic number field k as a function on the set of the finite primes of k with value in (𝔭/𝔭^{2}) ∪ {∞} for each prime 𝔭. According to his definition, the congruence

α^{N(𝔭)−1} ≡ 1 (mod 𝔭^{2})

is equivalent to dα(𝔭) = 0 for α with ord_{𝔭}α = 0, where N(𝔭) denotes the absolute norm of 𝔭 and ord_{𝔭} the normalized additive 𝔭-adic valuation. Thus, when l is a prime divisor of NH_{p}, p can be considered as a zero of the differential dl, and therefore we have got some big prime numbers l whose differential dl has a small zero. We do not know when NH_{p} has a big prime divisor, however, any other method of getting big prime numbers whose differential has a given small zero seems to be unknown.

References

1. W. L. Fouché, Arithmetic properties of Heilbronn sums, J. Number Theory 19 (1984), 1–6.

2. Y. Ihara, On Fermat quotient and “the differentials of numbers”, Algebraic analysis and number theory (Kyoto, 1992). Sūrikaisekikenkyūsho Kōkyūroku No. 810 (1992), 324–341. (Japanese); English translation by S. Hahn with supplement : the Univ. Georgia Preprint Series, No. 9, Vol. 2 (1994), 16 pp.

3. J. Suzuki, On the generalized Wieferich criteria, Proc. Japan Acad. Ser. A 70 (1994), 230–234.

Typeset by AMS-TEX

BY KEN YAMAMURA

4. A. Wieferich, Zum letzten Fermat’scen Theorem, J. reine angew. Math. 136 (1909), 293–302.

Table

p  NH_{p} (factorization)
3  −1
5  −1
7  97 (prime)
11  −243 = −3^{5}
13  12167 = 23^{3}
17  577 (prime)
19  221874931 (prime)
23  157112485811 (prime)
29  −2480435158303 = −137 · 18105366119
31  310695313260929 = 3727 · 34367 · 2425681
37  −51140551819476687829 (prime)
41  2727257042363914863401 = 17435281 · 156421742922521
43  −2572343484535669027372727 = −19^{4} · 19738518615846018887
47 1052824394331287344099620777449 = 53^{2} · 374803985165997630508942961

Continued

p  NH_{p} (factorization)
53  2487208425325253346238365486641837

= 15692059 · 158501088055127331998838743

59  −385919393422883989361433232474228039

= −53^{2} · 191559377 · 717201928003392149777423

61  2889586185058029549518039208456783772267

= 2281 · 5749 · 4431029 · 9582073 · 94339087 · 55012619117

67  −3332585486216383041159811020811948077704987791

(prime)

71  −159405892326070310729463014970352959169179259

= −11^{4} · 161323 · 498577 · 3607523 · 83966159 · 446880809176117

73  −53638311006466780577262790821628723384059340479403

= −523605697 · 102440273881257599413137765805532714408299

79  1040611956630950417455971245710339238106740379250667

= 31^{5} · 5872927 · 6189074918318922117260814024643164971

83  −36379133886955705100817312027432705145152593636880827

= −821 · 77069 · 2611457 · 37082337983851 · 5937170037444785228628289

89 78527589016099848753383963120976963786343522050332935396040538173

= 86955810683941·

903074658248253496993647901544985025793732684727353

Continued

p  NH_{p} (factorization)
97  698\

7145228295591002299725696981364375137003233828984685582960524329359233 = 107^{4}· 53304596405474189704771268696350123732349499857182084995688692433

101 205327420477226843839183486935873718130154715571336879842986005891861

= 181 · 37123837 · 7170291820167889761209· 4261658917838124892509819903468033757

103  −462533831800\

0359391437413980824742138776297993289056565984459759408380551737401739 = −43^{4} · 103387 · 13062349 · 2896538977081 · 25576909626197329· 13522438648797285977136546015865597

107  3182289\

1507341406977217349319487669649476442546560198075654864917635383601479 = 317 · 5573· 1801321915847158929132593963317259683743128487709738315574860139532331\ 9

109  −30\

3302263441082569473629038588196656024659930292341570668357212343800711 = −1053179481973 · 1856663508323066260741849· 155110093641829928440562381127274243

113  −7694239015349\

7767429914343980339482985073343801006021161946056411554553964025879073 = −373^{4} · 1901 · 83431 · 1876238197 · 878927113663 · 1479243444457147· 10274017271704880087861366539

127  474717820893761414463929\

5492364951760322843736468305200749920768316308728526708293193415249561 = 124693 · 989186813 · 4998938284506800044130459195709991663603· 7699054048370084195284674973132282435843

131  −4713205856787250624553328031322440\

4616730614746434844579685868984701459108116527816628360809843065647189 = −15527 · 4681575071378573167132041293231761269803· 648390829474870418054128536839110455896115986213993796049769

BY KEN YAMAMURA

Continued

p  NH_{p} (factorization)
137  −54195448488988962490696384958866927734769229610\

4110118512342156238930252641057216698614946914163629424067519146176693 = −19^{14} · 14813679214403· 4578776712575894828622557161712412748692380404006494845948685798378247\ 8878729044466111

139  −3800362258001898285462987682357007429335089\

7239701666267640012401085141966595733157507892775469822945274116456187 = −191336777 · 8956169923 · 262667607876648135474037816051211· 84430176767398056681786668521842356130098324476153501150369027

149  −12728259570934615961575936339731693311720285\

8698217643118069759540410288374132484347223198076781796628896036706953 = −2113032279103 · 971838776139621097101307· 1131231122299780402119817577743522037· 54792021516549081535015239045137510489089

151  −226220183752406717453240433137275823715353530524\

7601499841310573433625911083261391224513319972595087448647132163224213 = −751 · 485593 · 19521449 · 37271788723402129829165298935438630863· 8525634497904543561410673510415906089486140839369341104947753493

157  −32284141300994206874836369130115779567197863396965601717250\

6702102351499016839642651831758329191555792238365653679113136539279087 = −857 · 3843000354007 · 15521431265343739673297· 6315478006326723387129639712472083329659500466899459930082995404347174\ 673370592769632060929

163 −7400225900173044830752440964978020841650929721292858114533692410233\

4913764336703026458035333867875457182509765950769032048806745429876429 = −6728543 · 70039027 · 5307753301· 2958510478774496962321654194291483680592933584582335066954912538765536\ 3319446216503914460708865172851241892344989

167  −33648\

8141086228243405463024376740897618280866598765032176052042739901426421\ 1512093790822320811593498420307101467116666034986326092936688255357587 = −3329 · 8933 · 2196079 · 16450481 · 118968387511 · 8436118669266674509· 8106815585882319655197851· 384952836950146769193219378076614720765632935121532360773314388163841

Continued

p  NH_{p} (factorization)
173  235\

9888485036911676097144977103256615498605978952230357963959995333864898\ 5037911635912060569528415671016951895238565935417163603092974421811303 = 538068531941 · 2165178511215844193887· 498815472584183049034672308412437395777848829· 40608809983047366987458241147232962890167307939772842740885172521

179  −981129822\

9060755636390132454490199504549750209182495711548717749580196638567785\ 9016423790943993407508510771146277710692400032996008342011034607899411 = −9533 · 12391 · 29870689 · 174097380466511599 · 8039246656875405528031· 38181012604274678735504185933129· 520343644245128729336506518111110829044203868061631111031636033

181  834214044622\

0401236471710103774235552016889675552156737344104253271141364412906177\ 3024779605587918548945038460030735219485319390205304037820107811771421 = 313 · 15473 · 16334779 · 540663780732696860816693· 1950374876757737425205129479590276127371557587185955358112647879759569\ 307989337034398250610663295527459625340996307

191  588361853127797525355588431\

8450785465773956630782976103324422378319722195408700875770543712348086\ 4996882062236223115867826570351679728919288417113394050291284467379139 = 560039 · 1491421 · 2325901 · 420618014095412729· 7200238593297147058322107041201602006918465944294990750470416938817623\ 1825198157341482203535519045276248054109691892259110820011989

193  82021875512156306944935012226379791\

2751745910691684581444845017891688832181797728810320744108953927179129\ 2778246484950034509302296657355322636193771148520225262634144822830493 = 3996791· 2052193259846619624216903316345032584270095460812648400791647644051662\ 5267165526764104521455061171003916123655819093242640181341222639250645\ 42077324320566230574136637323

197  8100377444379552395200684\

9877013164569166155411997097852443460679367986816838277094280972497680\ 9448729950053905743939078758577271718917937609180623535983631421128689 = 22768151597· 3557766826116390775892234581075517272662252794819929793374322492630054\ 0925214341673588151529945623838862129037401978920017579030516102810434\ 444272359786837c

Continued

p  NH_{p} (factorization)
199  −792516883467304706673170988191762\

3064076503143780488085154184511373169271770279407843282307889741832791\ 7034578117614508480543956088680144983205923975063702143645452785379827 = −6813473 · 35597417· 3267544400866763276255671684878666952615472367764668340170715025221163\ 7618354175169538786738410118374901951379724078282820150618645694397046\ 1798427329808795547c

Hashirimizu Yokosuka 239-8686, Japan