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A348598 Least prime p of the form k^2+1 such that p == A002496(n) (mod A002496(n+1)) with p>A002496(n), or 0 if no such p exists.

%I A348598 #54 Jun 22 2022 09:25:22
%S A348598 17,1297,90001,1008017,147457,2421137,15952037,1378277,7203857,
%T A348598 107122501,164968337,34503877,38688401,4851958337,1075577617,
%U A348598 197121601,1044582401,315559697,70924211857,730296577,20705483237,15103426817,197740302401,4587352901,155964965777
%N A348598 Least prime p of the form k^2+1 such that p == A002496(n) (mod A002496(n+1)) with p>A002496(n), or 0 if no such p exists.
%C A348598 a(n) == 1, 5 (mod 16).
%C A348598 Conjecture: Consider the smallest prime p of the form k^2+1 such that p is congruent to A002496(n) modulo q, q prime of the form m^2+1 > A002496(n). Then q = A002496(n+1).
%C A348598 Corollary: For any pair (A002496(n), A002496(n+1)), there exist two integers m, k such that A002496(m) = A002496(n) + k*A002496(n+1), m>n+1 and n=1,2,3,...
%C A348598 Examples (see A352582):
%C A348598   A002496(3)   = A002496(1) +    3*A002496(2),
%C A348598   A002496(11)  = A002496(2) +   76*A002496(3),
%C A348598   A002496(49)  = A002496(3) + 2432*A002496(4),
%C A348598   A002496(113) = A002496(4) + 9980*A002496(5).
%H A348598 Michel Lagneau, <a href="/A348598/a348598.pdf">a(n),n=1..90</a>
%e A348598 a(2) = 1297 because 1297 == A002496(2) (mod A002496(3)) => 1297 == 5 (mod 17).
%p A348598 with(numtheory):T:=array(1..30000):k:=0:
%p A348598 nn:=500000:
%p A348598   for m from 1 to nn do:
%p A348598    if isprime(m^2+1)
%p A348598     then
%p A348598      k:=k+1:T[k]:=m^2+1:
%p A348598     else
%p A348598    fi:
%p A348598   od:
%p A348598   for n from 1 to 32 do:
%p A348598   ii:=0:r:=T[n]:q:=T[n+1]:
%p A348598    for i from 1 to k while(ii=0) do:
%p A348598      p:=T[i]:r1:=irem(p,q):
%p A348598         if r1=r and p>q
%p A348598          then
%p A348598          ii:=1: printf(`%d, `,p)
%p A348598          else
%p A348598         fi:
%p A348598        od:
%p A348598       od:
%Y A348598 Cf. A002496, A352582.
%K A348598 nonn
%O A348598 1,1
%A A348598 _Michel Lagneau_, Mar 20 2022