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A358313
Primes p such that 24*p is the difference of two squares of primes in three different ways.
1
5, 7, 13, 17, 23, 103, 6863, 7523, 11807, 11833, 22447, 91807, 100517, 144167, 204013, 221077, 478937, 531983, 571867, 752293, 1440253, 1647383, 1715717, 1727527, 1768667, 2193707, 2381963, 2539393, 2957237, 3215783, 3290647, 3873713, 4243997, 4512223, 4880963, 4895777, 5226107, 5345317, 5540063
OFFSET
1,1
COMMENTS
The positive integer solutions of 24*p = x^2 - y^2 are (x = p+6, y = p-6), (x = 2*p+3, y = 2*p - 3), (x = 3*p+2, y = 3*p-2) and (x = 6*p+1, y=6*p-1). Since at least one of these is always divisible by 7, it is impossible for 24*p to be the difference of two squares of primes in 4 different ways.
Primes p such that three of the pairs (p +- 6), (2*p +- 3), (3*p +- 2), (6*p +- 1) are pairs of primes.
Except for 5, all terms == 3 or 7 (mod 10).
LINKS
EXAMPLE
a(3) = 13 is a term because 13 is prime, 13 +- 6 = 19 and 7 are primes, 2*13 +- 3 = 29 and 23 are primes, and 3*13 +- 2 = 37 and 41 are primes.
MAPLE
filter:= proc(p) local t;
if not isprime(p) then return false fi;
t:= 0;
if isprime(p+6) and isprime(p-6) then t:= t+1 fi;
if isprime(2*p+3) and isprime(2*p-3) then t:= t+1 fi;
if isprime(3*p+2) and isprime(3*p-2) then t:= t+1 fi;
if isprime(6*p+1) and isprime(6*p-1) then t:= t+1 fi;
t = 3
end proc:
select(filter, [seq(i, i=3..10^7, 2)]);
CROSSREFS
Cf. A124865.
Sequence in context: A253297 A163385 A368277 * A288449 A178218 A314323
KEYWORD
nonn
AUTHOR
J. M. Bergot and Robert Israel, Nov 08 2022
STATUS
approved