A325437 - OEIS

A325437

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

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

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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 *)