A392469 Primes p that set records for the minimum k in the Erdős-Straus y = k*p construction for 4/p = 1/x + 1/y + 1/z.

73, 193, 1201, 2521, 3361, 33289, 90841, 144169, 167521, 225289, 361321, 915961, 954409, 1853329, 2031121, 605531161, 6736278481, 11602225441, 24707638729, 49563785881, 133050918961, 189241671529

Offset

1,1

Comments

For the Erdős-Straus conjecture, a standard approach for prime p sets y = k*p and searches for the minimum k such that a solution exists.

a(n) is the n-th prime that achieves a new record for minimum k among all smaller primes.

The corresponding record k-values form a companion sequence (submitted simultaneously).

All terms a(n) for n >= 15 satisfy: (2/p) = (3/p) = (5/p) = (7/p) = (11/p) = (13/p) = 1, where (a/p) is the Legendre symbol. These primes lie in residue classes mod 120120 = lcm(8,3,5,7,11,13).

For n >= 15, a(n) == r (mod 120120) where r is one of {961, 6241, 58969, 109201}.

Note dramatic "phase transitions": k jumps from 96 to 624 at a(15), and from 990 to 1484 at a(21).

The ratio k(p)/log(p) appears bounded, with maximum observed value 70.59 at a(22).

Verified by GPU-accelerated search to 10^13. Search ongoing as of Jan 2026.

Links

Table of n, a(n) for n=1..22.

Thomas Bloom, Erdős Problem #242

Wikipedia, Erdős-Straus conjecture.

Formula

Empirically, k(a(n)) < 71 * log(a(n)) for all known terms.

Example

a(1) = 73 requires k = 2.
a(15) = 2031121 requires k = 624, a dramatic jump from k = 96 at a(14).
a(22) = 189241671529 requires k = 1833, the current record.

Crossrefs

Cf. A073101, A075245, A075246, A075247, A075248, A192789, A287116.

Sequence in context: A142741 A088199 A140010 * A385094 A122723 A089786

Adjacent sequences: A392466 A392467 A392468 * A392470 A392471 A392472

Keyword

nonn,hard,more

Author

Jeffrey H. Gold, Jan 13 2026

Status

approved