A394632 Least prime q such that q^2 - n! is again the square of a prime, or 0 if no such prime exists.

0, 0, 0, 0, 7, 13, 29, 73, 223, 1201, 36313, 7109, 45341, 870517, 297377, 3240247, 6847843, 51948199, 80015629, 390403049, 1613360743, 22511523371, 34121600333, 162109085183, 949484701001, 5056628950459, 23744121396563, 2760884016693079, 562608177258653, 8107890641155673

Offset

0,5

Comments

Upon the observation that 7^2 - 5^2 = 49 - 25 = 24 = 4!, it is natural to ask whether there are primes p, q such that q^2 - p^2 = n! for all n >= 4.

If d1 * d2 = n! = q^2 - p^2 = (q + p) * (q - p) then we can set d1 = (q - p) and d2 = (q + p). Solving for p and q in terms of d1 and d2 gives p = abs((d1 - d2) / 2) and q = (d1 + d2)/2. As p and q are prime we must have gcd(d1, d2) = 2. - David A. Corneth, Apr 04 2026

Links

Table of n, a(n) for n=0..29.

David A. Corneth, PARI program

Formula

a(n) = sqrt(A394631(n)^2 + A000142(n)^2) = A000040(A394633(n)).

Example

See A394631.

Prog

(PARI) apply( {A394632(n, f=n!)=n>3&&forprime(p=3, oo, issquare(p^2+f, &q)&& isprime(q)&& return(q))}, [0..16]) /* For illustration, slow for n > 20. Better use: */
{A394632(n)=sqrtint(A394631(n)^2+n!)}  /* with more efficient code given there */
(PARI) \\ See Corneth link
(PARI) a(n)=if(n<4, return(0)); my(N=n!, L=List()); fordiv(vecprod(primes([3, n])), d, my(U=2*gcd(d^n, N)); my(p=(N/U+U)/2, q=abs(N/U-U)/2); if(ispseudoprime(q)&&ispseudoprime(p), listput(L, [p, q]))); vecsort(Vec(L))[1][1]; \\ Daniel Suteu, Apr 10 2026

Crossrefs

Cf. A001248 (squares of primes), A000142 (factorial).

Cf. A394631 (primes p = sqrt(a(n)^2 - n!)), A394633 (primepi(a(n))), A394629 (total number of primes {p, q} s.t. q^2 - p^2 = n!).

Sequence in context: A182624 A096333 A133325 * A336117 A381095 A283709

Adjacent sequences: A394629 A394630 A394631 * A394633 A394635 A394636

Keyword

nonn

Author

M. F. Hasler, Apr 04 2026

Extensions

More terms from David A. Corneth, Apr 04 2026

Status

approved