cp's OEIS Frontend

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

%I A236767 #22 Jan 24 2022 08:03:01
%S A236767 2,10,37,82,442,577,730,901,1090,1297,1765,2026,4357,5185,5626,7570,
%T A236767 8650,9217,9802,10405,11026,15130,17425,18226,23410,24337,26245,31330,
%U A236767 34597,35722,40402,41617,47962
%N A236767 Numbers whose square is a fourth power plus a prime.
%C A236767 Based on a 1999 observation of Alessandro Zaccagnini (via John Robertson) intended to dissuade expectation of a finite fourth-power analogy to A020495, A045911.
%C A236767 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.
%H A236767 Hans Havermann, <a href="/A236767/b236767.txt">Table of n, a(n) for n = 1..1000</a>
%H A236767 John Robertson, <a href="http://mathforum.org/kb/message.jspa?messageID=1676582">Integers of the form x^2+kp</a> (see last paragraph)
%F A236767 A089001^2 + 1
%e A236767 2 is a term because 2^2 = 1^4 + 3;
%e A236767 10 is a term because 10^2 = 3^4 + 19;
%e A236767 37 is a term because 37^2 = 6^4 + 73.
%t A236767 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 *)
%t A236767 s=A089001; s^2+1 (* based on formula *)
%Y A236767 Cf. A020495, A045911, A089001.
%K A236767 nonn
%O A236767 1,1
%A A236767 _Hans Havermann_, Jan 30 2014