cp's OEIS Frontend

A325437 Final digit of primes of the form k^2 + 1.

%I A325437 #21 Feb 16 2025 08:33:58
%S A325437 2,5,7,7,1,7,7,1,7,7,7,1,7,7,7,7,7,1,7,1,7,1,7,7,1,7,7,1,7,1,1,7,1,7,
%T A325437 7,7,1,7,1,7,1,1,7,1,7,1,1,7,1,7,7,7,1,1,7,7,7,1,7,1,1,7,1,7,7,7,1,7,
%U A325437 1,7,7,7,7,1,7,7,7,7,7,7,7,7,7,1,7,1,7
%N A325437 Final digit of primes of the form k^2 + 1.
%C A325437 This sequence is presumably infinite. See 1st comment of A002496.
%C A325437 For k > 2, i.e., primes > 5 the final digit is always 1 or 7. Proof: Let k = 2*m - 1 odd. Then k^2 + 1 is divisible by 2, hence prime only for m = 1. Let k = 2*m even. Then k^2 + 1 = 4*m^2 + 1. The final digit of multiples of four is 4, 8, 2, 6, 0, 4, 8, 2, 6, 0, ... and of squares 1, 4, 9, 6, 5, 6, 9, 4, 1, 0, ... (cf. A008959), hence the last digit of the product 4*m^2 is 4, 6, 6, 4, 0, ... or of the sum 4*m^2 + 1 is 5, 7, 7, 5, 1, ... (cf. A053755) and therefore for primes > 5 the final digit is 1 or 7.
%C A325437 Accordingly, for large k approximately one-third of the primes of the form k^2 + 1 end in 1, two-thirds end in 7.
%H A325437 Robert Israel, <a href="/A325437/b325437.txt">Table of n, a(n) for n = 1..10000</a>
%H A325437 Edmund Landau, <a href="http://gdz.sub.uni-goettingen.de/dms/resolveppn/?PPN=GDZPPN002123789">Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion</a>, Jahresbericht der Deutschen Mathematiker-Vereinigung (1912), Vol. 21, page 208-228, here p. 224.
%H A325437 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LandausProblems.html">Landau's Problems.</a>, Nr. 4.
%H A325437 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Near-SquarePrime.html">Near-Square Prime.</a>
%F A325437 a(n) = A002496(n) mod 10.
%p A325437 seq(k mod 10,k=select(isprime,[2,seq(4*i^2+1,i=1..10000)]));
%t A325437 Mod[#,10]&/@Select[Range[1000]^2+1,PrimeQ] (* _Harvey P. Dale_, Jul 05 2023 *)
%o A325437 (PARI) lista(nn) = {forprime(p=2, nn, if (issquare(p-1), print1(p % 10, ", ")););} \\ _Michel Marcus_, May 07 2019
%Y A325437 Cf. A001912, A002496, A005574.
%K A325437 nonn,base
%O A325437 1,1
%A A325437 _Martin Renner_, Apr 27 2019