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URL: https://oeis.org/A392479

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A392479
Pairs of prime numbers (p,q), p minimal, such that prime(n) = p^3 - 2*q, or (0,0) if no such pair exists.
1
2, 3, 5, 61, 3, 11, 5, 59, 13, 1093, 3, 7, 3, 5, 5, 53, 3, 2, 7, 157, 5, 47, 11, 647, 7, 151, 5, 41, 37, 25303, 31, 14869, 13, 1069, 23, 6053, 5, 29, 13, 1063, 23, 6047, 5, 23, 421, 37309189, 7, 127, 11, 617, 43, 39703, 5, 11, 61, 113437, 23, 6029, 19, 3373, 17, 2393
OFFSET
1,1
COMMENTS
Conjecture: Every prime number is of the form p^3 - 2*q for some primes p and q.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..20000 (the first 10000 pairs)
EXAMPLE
*---*-----------*--------------------*
| n | (p,q) | p^3-2*q = prime(n) |
*---*-----------*--------------------*
| 1 | (2,3) | 2 |
| 2 | (5,61) | 3 |
| 3 | (3,11) | 5 |
| 4 | (5,59) | 7 |
| 5 | (13,1093) | 11 |
| 6 | (3,7) | 13 |
For some pairs p is larger than q: 7^3 - 2*3 = 337 = prime(68).
MAPLE
T:= proc(n) option remember; local r, p;
r:= ithprime(n); p:= nextprime(iroot(r, 3)-2);
while not (h-> h>0 and h::even and isprime(h/2))(p^3-r)
do p:= nextprime(p) od; p, (p^3-r)/2
end:
seq(T(n), n=1..31); # Alois P. Heinz, Jan 13 2026
CROSSREFS
KEYWORD
nonn,look,tabf
AUTHOR
Michel Lagneau, Jan 13 2026
STATUS
approved