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A187401 Numbers k such that k^2 + 1 = p*q, p and q primes and |p-q| is square.

%I A187401 #25 Mar 07 2020 12:24:48
%S A187401 30,100,144,274,484,516,526,756,1046,1250,1714,1806,1834,2284,2440,
%T A187401 2610,2940,3524,3824,4190,5084,5746,6766,7486,9746,9920,10310,13024,
%U A187401 13210,15396,16916,17546,18726,19256,20000,21194,23214,24964,30370,30394,31126,31496,35180,36680,37816
%N A187401 Numbers k such that k^2 + 1 = p*q, p and q primes and |p-q| is square.
%C A187401 Note that if k^2+1 = p*q, then p+q cannot be a square. Proof by contradiction. There are two cases: p an odd prime and p=2. Case 1: suppose p and q are odd primes and q = y^2-p. Note that y must be an even number in order for q to be odd. Then p(y^2-p) = x^2+1 for some even x. Rearranging terms, we obtain p*y^2-1 = p^2+x^2. Looking at this equation modulo 4, we obtain -1 = 1, a contradiction. Case 2: Let p=2. Then we obtain 2y^2-x^2 = 5, which has no solutions in integers. - _T. D. Noe_, Mar 10 2011
%H A187401 Robert Israel, <a href="/A187401/b187401.txt">Table of n, a(n) for n = 1..600</a>
%e A187401 20000 is in the sequence because 20000^2+1 = 19801 * 20201 and 20201 - 19801 = 20^2.
%p A187401 with(numtheory):nn:=50000:for i from 1 to nn do: n:=i^2+1:x:=factorset(n):x1:=nops(x):x2:=bigomega(n):if  x1=2 and x2=2 then z:=x[2]-x[1] :w:=sqrt(z):if w= floor(w) then printf(`%d,  `, i):else fi:else fi :od:
%p A187401 # Alternative:
%p A187401 N:= 500: # to get a(1) to a(N)
%p A187401 count:= 0:
%p A187401 for k from 2 by 2 while count < N do
%p A187401   f:= ifactors(k^2+1)[2];
%p A187401   if nops(f) = 2 and {f[1,2],f[2,2]}={1} and issqr(abs(f[1,1]-f[2,1])) then
%p A187401     count:= count+1;
%p A187401     A[count]:= k;
%p A187401   fi
%p A187401 od:
%p A187401 seq(A[i],i=1..count); # _Robert Israel_, Jun 09 2014
%t A187401 okQ[k_] := Module[{ff = FactorInteger[k^2+1]}, Length[ff] == 2 && ff[[All, 2]] == {1, 1} && IntegerQ[Sqrt[ff[[2, 1]] - ff[[1, 1]]]]];
%t A187401 Select[Range[2, 40000, 2], okQ] (* _Jean-François Alcover_, Jun 25 2019 *)
%o A187401 (Sage)
%o A187401 A = []
%o A187401 for k in range(2, 2000, 2):
%o A187401     K = k^2 + 1
%o A187401     f = prime_divisors(K)
%o A187401     if len(f) == 2:
%o A187401         if mul(f) == K:
%o A187401             if is_square(abs(f[0]-f[1])):
%o A187401                 A.append(k)
%o A187401 print(A) # _Peter Luschny_, Jun 10 2014
%Y A187401 Cf. A134406, A134407, A002522, A005574.
%K A187401 nonn
%O A187401 1,1
%A A187401 _Michel Lagneau_, Mar 09 2011