A083845 a(n)^2 + 1 is largest prime of the form x^2 + 1 <= 10^n.

2, 6, 26, 94, 314, 986, 3160, 9990, 31614, 99996, 316206, 999960, 3162246, 9999960, 31622764, 99999966, 316227734, 999999924, 3162277654, 9999999956, 31622776500, 99999999964, 316227766006, 999999999886, 3162277660140

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 never becomes a constant, but this has not been proved.

The ratio a(n+2)/a(n) appears to approach 10, as one might expect. - Bill McEachen, Nov 03 2013

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, A083846, A083847, A083848, A083849.

Programs

Mathematica

Do[ k = Floor[ Sqrt[ 10^n] - 1]; While[ !PrimeQ[k^2 + 1], k-- ]; Print[k], {n, 1, 25}]

Extensions

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