A299118 Squares s such that prime(s) + 2 is a square.

1, 4, 9, 7745089

Offset

1,2

Comments

Primes corresponding to the first four squares are 2, 7, 23, and 136866599. The sequence may be finite.

There may be no square s such that prime(s) + 1 is square (none was found up to 10^9).

This is a Diophantine problem of the form f(n^2) + A = m^2, where f(x) = prime(x), and the simplest case of A = 1 has probably no solutions unlike the same case with f(x) = primepi(x) that may even have an infinite number of solutions.

Links

Table of n, a(n) for n=1..4.

Example

prime(4) + 2 = 7 + 2 = 9, and both 4 and 9 are squares.

Mathematica

Select[Range[10^4]^2, IntegerQ@Sqrt[Prime[#] + 2] &]

Prog

(PARI) for(n=1, 10^4, issquare(prime(n^2)+2)&&print1(n^2 ", "))

Crossrefs

Cf. A000290 (squares), A000040 (primes), A011757 (primes with square indices).

Sequence in context: A260305 A229338 A111443 * A276272 A331450 A051499

Adjacent sequences: A299115 A299116 A299117 * A299119 A299120 A299121

Keyword

nonn,more

Author

Waldemar Puszkarz, Feb 02 2018

Status

approved