A236767 Numbers whose square is a fourth power plus a prime.

2, 10, 37, 82, 442, 577, 730, 901, 1090, 1297, 1765, 2026, 4357, 5185, 5626, 7570, 8650, 9217, 9802, 10405, 11026, 15130, 17425, 18226, 23410, 24337, 26245, 31330, 34597, 35722, 40402, 41617, 47962

Offset: 1

Author

Hans Havermann, Jan 30 2014

Keywords

• nonn

Comments

Based on a 1999 observation of Alessandro Zaccagnini (via John Robertson) intended to dissuade expectation of a finite fourth-power analogy to A020495, A045911.

It can be shown that A089001^2 + 1 are members of this sequence. David Applegate shows that they are the only members: If x^2 = y^4 + p, let a = x - y^2. Then y^4 + p = x^2 = (y^2 + a)^2 = y^4 + 2a*y^2 + a^2, so p = 2a*y^2 + a^2, and so a divides p. Since p is a prime, a must be a unit (that is, +1 or -1). But since p >= 2, a must be +1.

Examples

			2 is a term because 2^2 = 1^4 + 3;
10 is a term because 10^2 = 3^4 + 19;
37 is a term because 37^2 = 6^4 + 73.
		

Links

• Hans Havermann, Table of n, a(n) for n = 1..1000
• John Robertson, Integers of the form x^2+kp (see last paragraph)

Crossrefs

Cf. A020495, A045911, A089001.

Programs

Mathematica

r=Range[10000]^4; j=1; Do[c=i^2; k=c^2-Take[r,i]; Do[c++; j=j+2; k=k+j; If[MemberQ[PrimeQ[k], True], Print[c]], {2*i+1}], {i, 10000}] (* brute force *)
s=A089001; s^2+1 (* based on formula *)

Formula

A089001^2 + 1