A350544 a(n) is the least prime p such that there exists a prime q with p^2 + n = (n+1)*q^2, or 0 if there is no such p.
7, 5, 0, 11, 7, 13, 5, 0, 41, 23, 17, 10496997797584752004430879, 41, 11, 7
Offset: 1
Examples
a(3) = 0 as the only positive integer solution of p^2 + 3 = 4*q^2 is p=1, q=1, and 1 is not prime. a(4) = 11 as 11^2 + 4 = 125 = (4+1)*5^2 with 11 and 5 prime.
Links
- Robert Israel, Table of n, a(n) for n = 1 .. 179 with conjectured 0 values as -1.
Programs
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Maple
# Returned values of -1 indicate that either a(n) = 0 or a(n) > 10^1000. f:= proc(n) local m,x,y,S,cf,i,c,a,b,A,M,Sp; m:= n+1; if issqr(m) then S:= [isolve(x^2+n=m*y^2)]; S:= map(t -> subs(t,[x,y]),S); S:= select(t -> andmap(isprime,t),S); if S = [] then return 0 else return min(map(t -> t[1],S)) fi; fi; cf:= NumberTheory:-ContinuedFraction(sqrt(m)); for i from 1 do c:= Convergent(cf,i); if numer(c)^2 - m*denom(c)^2 = 1 then break fi od; a:= numer(c); b:= denom(c); A:= <|
>; M:= floor(sqrt(n)*(1+sqrt(a+b*sqrt(m)))/(2*sqrt(m))); S:= select(t -> issqr(m*t^2-m+1), [$0..M]); S:= select(t -> igcd(t[1],t[2])=1,map(t -> , S)); S:= map(t -> (t, <-t[1],t[2]>), S); if nops(S) = 0 then return 0 fi; for i from 0 do Sp:= select(t -> isprime(t[1]) and isprime(t[2]),S); if nops(Sp)>0 then return min(map(t -> t[1],Sp)) fi; S:= map(t -> A.t,S); if min(map(t -> t[1],S))>10^1000 then break fi; od; -1 end proc: map(f, [$1..20]);
Formula
a(n)^2 + n = (n+1)*A350550(n)^2 if a(n) > 0.
Comments