New Types of Congruences Involving Bernoulli Numbers and Fermat's Quotient | Zendy

H. S. Vandiver | Zendy

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New Types of Congruences Involving Bernoulli Numbers and Fermat's Quotient

Author(s) -

H. S. Vandiver

Publication year - 1948

Publication title -

proceedings of the national academy of sciences

Language(s) - English

Resource type - Journals

SCImago Journal Rank - 5.011

H-Index - 771

eISSN - 1091-6490

pISSN - 0027-8424

DOI - 10.1073/pnas.34.3.103

Subject(s) - bernoulli's principle , quotient , symmetry (geometry) , congruence relation , interpretation (philosophy) , pure mathematics , mathematics , quantum , bernoulli number , theoretical physics , fermat's last theorem , physics , quantum mechanics , computer science , geometry , thermodynamics , programming language

make Mp prime are 2, 3, 5, 7, 13, 17, 19, 31, 67, 127 and 257. Comparison of this list with the correct data recorded in the top line of the table presented below shows that Mersenne made five mistakes. p = 67 and 257 do not yield prime values for Mp, and p = 61, 89 and 107 were not included in his list of special primes. With reference to explicit factoring, attention should be called to a valuable paper4 by Professor D. H. Lehmer entitled "On the Factors of 2n 1." His investigations on 76 numbers unveiled eleven factors which fall within Mersenne's range. Incidentally two of his new factors confirmed the present writer's final residues for M167 and M229.

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