A078668 Primes that are the difference between two powers: y^z - x^z = prime.

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 127, 211, 271, 331, 397, 547, 631, 919, 1657, 1801, 1951, 2269, 2437, 4651, 8191, 14197, 61051, 131071, 371281, 524287, 543607, 723901, 1273609, 1803001, 2685817, 2861461, 5217031

Offset

1,1

Comments

This sequence is much faster to compute than y^z+x^z since the occurrence of a prime for y^z - x^z only takes place when z = 1 or y-x = 1. This is true because y-x or y+x divides y^z - x^z. Comment out the print(x" "y" "z" "v" "ct); in the script to avoid listing the detail to the screen. Sum of the reciprocals of Seq(20,130) converges to 1.6359026039776143431548856889889230600448729878668090784647536941979 31129745142466816093140975967179... to 100 places.

Is this sequence well-defined? The data could not be reproduced based on the name, comments, or existing code. Note that all odd primes can be represented as the difference of two squares so, for example, why is 67 = 34^2-33^2 not in the data. Restricting z>2, also does not reproduce this sequence. - Sean A. Irvine, Jul 11 2025

Links

Table of n, a(n) for n=1..45.

Formula

Seq(n, m) = y^z - x^z = p. x=1..n, y=x..n, p=1..m. Include if p is prime.

Example

919 = 18^3 - 17^3. 919 is prime.

Prog

(PARI) powerp(n, p) = { ct=0; sr=0; a=vector(n*n*p); for(x=1, n, for(y=x, n, for(z = 1, p, if(y-x == 1 || z == 1, v = y^z - x^z; if(isprime(v), ct+=1; a[ct] = v; /* print(x" "y" "z" "v" "ct); */ ); ); ); ); ); for(j=1, ct, for(k=j+1, ct, if(a[j] > a[k], tmp=a[k]; a[k]=a[j]; a[j]=tmp); ); ); for(j=1, ct, if(a[j]<>a[j+1], sr+=1.0/a[j]; print1(a[j], ", ")); ); print(); print(sr); }

Crossrefs

Sequence in context: A118850 A322443 A219697 * A386529 A038614 A386324

Adjacent sequences: A078665 A078666 A078667 * A078669 A078670 A078671

Keyword

easy,nonn,obsc

Author

Cino Hilliard, Dec 16 2002

Status

approved