Prime numbers seem to be distributed among the natural numbers with no law other than that of chance; however, their global distribution presents a quite remarkable smoothness. Such interplay between randomness and regularity has motivated scientists across the ages to search for local and global patterns in this distribution that could eventually shed light on the ultimate nature of primes. In this paper, we show that a generalization of the well-known first-digit Benford's law, which addresses the rate of appearance of a given leading digit d in datasets, describes with astonishing precision the statistical distribution of leading digits in the prime number sequence. Moreover, a reciprocal version of this pattern also takes place in the sequence of the non-trivial Riemann zeta zeros. We prove that the prime number theorem is, in the final analysis, responsible for these patterns.

The first-digit frequencies of prime numbers and Riemann zeta zeros

https://www.ems-ph.org/journals/show_pdf.php?issn=0013-6018&vol=67&iss=4&rank=4

Divisibility of power sums and the generalized Erdős–Moser equation

A classical result of Sylvester [21] (see also [16], [17]), generalizing Bertrand’s Postulate, states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k. More recent work in this vein, well surveyed in [18], has focussed on sharpening Sylvester’s theorem, or upon providing lower bounds for the number of prime divisors of such a product. As noted in [18], a basic technique in these arguments is to make a careful distinction between “small” and “large” primes, and then apply sophisticated results from multiplicative number theory. Along these lines, if we write ( n k ) = U · V, n ≥ 2k,

/pdf/on-the-factorization-of-consecutive-integers-5dpfp06l7j.pdf

On the factorization of consecutive integers

Let n > 1 be an odd composite integer. Write n - 1 = 2 s d with d odd. If either b d ≡ 1 mod n or b 2r d ≡ -1 mod n for some r = 0,1,..., s - 1, then we say that n is a strong pseudoprime to base b, or spsp(b) for short. Define ψ t to be the smallest strong pseudoprime to all the first t prime bases. If we know the exact value of ψ t , we will have, for integers n 10 36 ; and to give reasons and numerical evidence of K2- and C 3 -spsp's 12, where ψ' t (resp. ψ" t ) is the smallest K2- (resp. C 3 -) strong pseudoprime to all the first t prime bases. For this purpose we describe procedures for computing and enumerating the two kinds of spsp's < 10 36 to the first 9 prime bases. The entire calculation took about 4000 hours on a PC Pentium IV/1.8GHz. (Recall that a K2-spsp is an spsp of the form: n = pq with p, q primes and q - 1 = 2(p - 1); and that a C 3 -spsp is an spsp and a Carmichael number of the form: n = q 1 q 2 q 3 with each prime factor q i = 3 mod 4.).

https://www.ams.org/mcom/2007-76-260/S0025-5718-07-01977-1/S0025-5718-07-01977-1.pdf

Two kinds of strong pseudoprimes up to 10^{36}

/pdf/a-note-on-jesmanowicz-conjecture-concerning-primitive-34z9jz8lu8.pdf

A note on Jeśmanowicz' conjecture concerning primitive Pythagorean triplets

references. The index of notations used in the text fills six pages. There are 29 tables plus other lists. There are however no graphs or other illustrations. A plot of each index (or the number of digits) of the Mersenne primes on a logarithmic scale against their dates of discovery would show the significance of GIMPS (Great Internet Mersenne Prime Search) since 1996, which is clear from the table. But what is not easily seen from the table is the increase in the rate of finding these large primes in this period. In 1983, the year when the 29th Mersenne prime was found, I predicted from such a plot, and assuming no changes in the trends for resources or methods, that the 40th Mersenne prime would be found in 2008 ± 3 years [1]. It was found in November 2003 and confirmed soon afterwards, over a year before my window, due to GIMPS. A similar plot for the largest known primes during the last 30 years can be seen at www.utm.edu/research/primes/notes/by_year.html.

P. Shiu

Papers

Unsolved problems in number theory (3rd edn), by Richard K. Guy. Pp. 437. £54.00. 2004. ISBN 0 387 20860 (Springer-Verlag).