A valuable memory

This paper by the legendary N. J. A. Sloane mentioning the result on prime sequences (particularly GPF-Fibonacci sequences and a conjecture on GPF-Tribonacci) that I got in collaboration with my student Greg Back meant a lot to me. It is, for me, a most valuable "Math memory".

 

On the Liouville function at polynomial arguments - Joni Teräväinen

 

Integer partitions detect the primes

A remarkable new paper:

Integer partitions detect the primes, by William Craig, Jan-Willem van Ittersum, Ken Ono

https://arxiv.org/abs/2405.06451

The Riemann Hypothesis, Explained - Quanta Magazine

 

Peter Sarnak on Möbius Disjointness

The Patterns in the Primes, with Andrew Granville

 

Mobius Randomness and Dynamics - Peter Sarnak

First group of 6 consecutive integers with exactly 6 prime factors each

 

Stirring the pot

 

On a related note, a page on patterns with primes, with contributors including Charles W. Trigg and Martin Gardner:

PRIME PATTERNS

The attractiveness of primes for experimental mathematics...

'Although the prime numbers are rigidly determined, they somehow feel like experimental data'

Timothy Gowers. Mathematics: A Very Short Introduction (Oxford Univ. Press, 2002) p.118; as cited in "surprising connections between number theory and physics"

All primes in terms of one: non-associative algebra and Google cloud computing

Under addition, the positive integers 1, 1+1, 1+1+1,.... form a cyclic (semigroup) structure generated by 1. We will explore a way of representing prime numbers in terms of the single generator g = 2, under a non-associative, non-commutative binary operation "o" on the set A of all primes, defined by setting x o y:=P(2x+y) where P is the greatest prime factor function. For example, 3 = 2 o 2, 5 = 2 o (2 o (2 o (2 o 2)))), etc. In a joint work with Paul A. Vicol from Simon Fraser University, we managed to verify that all primes up to 7259167 can be expressed as non-associative products involving the symbols 2, o, and parentheses ),(. This computational evidence points towards a cyclicity conjecture for the (magma) structure (A, o). Moreover, we searched for other similar non-associative algebraic structures on A (prime magmas) that might be cyclic, established a set of fairly restrictive necessary conditions for cyclicity, formulated a more general cyclicity conjecture for special prime magmas, and found computational evidence (after days of running Julia programs on a GCE platform) for the cyclicity of the structures in a representative set. (October 24, 2014 - MAA Ohio Fall Meeting)

A major breakthrough on primes: Yitang Zhang

Bounded gaps between primes - by Yitang Zhang - Ann. Math. (forthcoming)