A083846 a(n) is the largest prime of the form x^2 + 1 <= 10^n.

5, 37, 677, 8837, 98597, 972197, 9985601, 99800101, 999444997, 9999200017, 99986234437, 999920001601, 9999799764517, 99999200001601, 999999202999697, 9999993200001157, 99999979750774757, 999999848000005777

Offset: 1

Author

Harry J. Smith, May 05 2003

Keywords

• nonn

Comments

It is conjectured that the number of primes of the form x^2+1 is infinite and thus this sequence does not become a constant, but this has not been proved. It is easily shown that all terms greater than 5 end in 1 or 7.

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, th. 17.
• P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 190.

Links

• Eric Weisstein's World of Mathematics, Landau's Problems.

Crossrefs

Cf. A005574, A002496, A083844, A083845, A083847, A083848, A083849.

Programs

Mathematica

Do[ k = Floor[ Sqrt[ 10^n] - 1]; While[ !PrimeQ[k^2 + 1], k-- ]; Print[k^2 + 1], {n, 1, 19}]
lpf[n_]:=Module[{p=NextPrime[10^n,-1]},While[!IntegerQ[Sqrt[p-1]],p= NextPrime[ p,-1]];p]; Array[lpf,10] (* The program generates the first 10 terms of the sequence. *) (* Harvey P. Dale, Feb 11 2023 *)

Extensions

Edited and extended by Robert G. Wilson v, May 08 2003