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 A325437 Final digit of primes of the form k^2 + 1. 3
 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, 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, 1, 7, 7, 7, 7, 1, 7, 7, 7, 7, 7, 7, 7, 7, 7, 1, 7, 1, 7 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS This sequence is presumably infinite. See 1st comment of A002496. 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. Accordingly, for large k approximately one-third of the primes of the form k^2 + 1 end in 1, two-thirds end in 7. LINKS Robert Israel, Table of n, a(n) for n = 1..10000 Edmund Landau, Gelöste und ungelöste Probleme aus der Theorie der Primzahlverteilung und der Riemannschen Zetafunktion, Jahresbericht der Deutschen Mathematiker-Vereinigung (1912), Vol. 21, page 208-228, here p. 224. Eric Weisstein's World of Mathematics, Landau's Problems., Nr. 4. Eric Weisstein's World of Mathematics, Near-Square Prime. FORMULA a(n) = A002496(n) mod 10. MAPLE seq(k mod 10, k=select(isprime, [2, seq(4*i^2+1, i=1..10000)])); MATHEMATICA Mod[#, 10]&/@Select[Range[1000]^2+1, PrimeQ] (* Harvey P. Dale, Jul 05 2023 *) PROG (PARI) lista(nn) = {forprime(p=2, nn, if (issquare(p-1), print1(p % 10, ", ")); ); } \\ Michel Marcus, May 07 2019 CROSSREFS Cf. A001912, A002496, A005574. Sequence in context: A290410 A078320 A306992 * A141430 A021392 A131688 Adjacent sequences: A325434 A325435 A325436 * A325438 A325439 A325440 KEYWORD nonn,base AUTHOR Martin Renner, Apr 27 2019 STATUS approved

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