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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
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
Sequence in context: A290410 A078320 A306992 * A141430 A021392 A131688
KEYWORD
nonn,base
AUTHOR
Martin Renner, Apr 27 2019
STATUS
approved

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Last modified June 20 12:17 EDT 2024. Contains 373527 sequences. (Running on oeis4.)