Irregular primes and cyclotomic invariants
Irregular primes and cyclotomic invariants
复制标题
不规则素数和分圆不变量
DOI：
10.1090/s0025-5718-1975-0376606-9
复制
发表时间：
1975
影响因子：
2
通讯作者：
W. Johnson
中科院分区：
文献类型：
--
作者：
W. Johnson
已计算出小于30000的不规则素数表，并存入UMT文件。不规则素数在这个范围内的比例为0.3924，接近于启发式预测的1 -e 112。对于所有素数指数p < 30000，证明了费马大定理，并对这些素数完全确定了Iwasawa的环切不变量p、Xp和vp。计算表明，对于这个范围内的p， A,p = 0，不变量Xp和vp都等于p. 1的不规则性指标。历史总结与导论。自从Kummer证明了他的不朽定理，即费马方程xP + yP = ZP对于正则素数指数p没有非平凡积分解以来，大约已经125年了。如果一个素数p不除任何伯努利数B2， B4, * *， Bp_3的分子，则称为正则素数p。这个条件等价于p不除将一个单位的原始p次根邻接到有理域所得到的环切场的类数的假设。这些结果的详细说明出现在2010年，Kummer自己开始寻找他的费马猜想的证明不适用于的素数。到1874年，他已经确定了小于165的37个奇数素数中有8个是不规则的，包括素数157，它是第一个能整除两个伯努利数分子的素数。在桌上计算器的帮助下，Vandiver和他的同事们继续计算到20世纪30年代的617。到1955年，Vandiver， D. H. Lehmer， E. Lehmer， Selfridge和Nicol [12]， [24]， b[17]在洛杉矶的SWAC计算机上完成了4001的计算。1963年，D. H. Lehmer[11]报告了统计结果到10000,1964年，Selfridge和Pollack[11]宣布在UCLA的IBM 7090上完成了到25000的表。后面这些表格还没有印出来。在1970年，Kobelev[10]发布了表到5500，并在1973年被作者[8]扩展到8000。1974年6月13日收。AMS （MOS）学科分类（1970年）。主10A40, 12A35, 12AS0。
The table of irregular primes less than 30000 has been computed and deposited in the UMT file. The fraction of irregular primes in this range is 0.3924, close to the heuristic prediction of 1 -e 112. Fermat's Last Theorem has been verified for all prime exponents p < 30000, and the cyclotomic invariants ,p, Xp, and vp of Iwasawa have been completely determined for these primes. The computations show that for p in this range, A,p = 0 and the invariants Xp and vp both equal the index of irregularity of p. 1. Historical Summary and Introduction. It has been roughly 125 years since Kummer proved his monumental theorem that the Fermat equation xP + yP = ZP has no nontrivial integral solutions for regular prime exponents p. A prime p is called regular if it does not divide the numerator of any of the Bernoulli numbers B2, B4, * * , Bp_3. This condition is equivalent to the assumption that p does not divide the class number of the cyclotomic field obtained by adjoining a primitive pth root of unity to the rational field. A detailed exposition of these results appears in [1]Kummer himself began the search for the primes to which his proof of the Fermat conjecture did not apply. By 1874 he had determined that exactly 8 of the 37 odd primes less than 165 are irregular, including the prime 157, the first prime which divides the numerators of two of the Bernoulli numbers in question. With the aid of desk calculators, Vandiver and his associates [23] continued the computations to 617 in the 1930's. By 1955, Vandiver, D. H. Lehmer, E. Lehmer, Selfridge and Nicol [12], [24], [17] had completed the computations to 4001 on the SWAC computer at Los Angeles. In 1963, D. H. Lehmer [11] reported statistical results to 10000, and in 1964 Selfridge and Pollack [18] announced completion of the table to 25000 on the IBM 7090 at UCLA. These latter tables have not appeared in print. In 1970, Kobelev [10] published the table to 5500 and this was extended to 8000 by the author [8] in 1973. Received June 13, 1974. AMS (MOS) subject classifications (1970). Primary 10A40, 12A35, 12AS0.