A368418 Numbers X such that X^2 + Y^2 = 10^(2*k) + 1, with X > Y > 0 and k is the decimal digit length of X-1.

10, 76, 100, 980, 1000, 8824, 10000, 76249, 87551, 98020, 100000, 753424, 766424, 999800, 1000000, 7209049, 7241380, 8220640, 8463640, 9801980, 9879740, 9990280, 10000000, 77053825, 78173720, 80404255, 83754376, 84711551, 86600176, 90880001, 93094625, 93728480

Offset

1,1

Comments

The values X and Y are used in finding A368416.

The number of terms for a given k is 2^(f-1), where f = A119704(2*k) is the number of prime factors of 10^(2*k) + 1.

References

Frits Beukers, "Getallen - Een inleiding" (In Dutch), Epsilon Uitgaven, Amsterdam (2015).

Links

A.H.M. Smeets, Table of n, a(n) for n = 1..67

T. Granlund, Factors of 10^n + 1.

Alf van der Poorten, The Hermite-Serret Algorithm and 12^2 + 33^2. In: Lam, KY., Shparlinski, I., Wang, H., Xing, C. (eds) Cryptography and Computational Number Theory. Progress in Computer Science and Applied Logic, vol 20. Birkhäuser, Basel.

Example

10 is a term since X = 10, Y = 1, k = 1 and 10^2 + 1^2 = 101.
76 is a term since X = 76, Y = 65, k = 2 and 76^2 + 65^2 = 10001.
980 is a term since X = 980, Y = 199, k = 3 and 980^2 + 199^2 = 1000001.

Crossrefs

Cf. A098608, A119704, A368416.

Sequence in context: A228416 A049392 A136869 * A108277 A061319 A223994

Adjacent sequences: A368415 A368416 A368417 * A368419 A368420 A368421

Keyword

nonn,base

Author

A.H.M. Smeets, Dec 24 2023

Status

approved