A283395 Squarefree numbers m congruent to 1 modulo 4 such that the fundamental unit of the field Q(sqrt(m)) has the form x+y*sqrt(m) with x, y integers.

17, 33, 37, 41, 57, 65, 73, 89, 97, 101, 105, 113, 129, 137, 141, 145, 161, 177, 185, 193, 197, 201, 209, 217, 233, 241, 249, 257, 265, 269, 273, 281, 305, 313, 321, 329, 337, 345, 349, 353, 373, 377, 381, 385, 389, 393, 401, 409, 417, 433, 449, 457, 465, 473, 481, 485, 489, 497, 505, 521, 537, 545, 553, 557, 561, 569, 573

Offset: 1

Author

Emmanuel Vantieghem, Mar 07 2017

Keywords

• nonn

Comments

Squarefree integers m congruent to 1 modulo 4 such that the minimal solution of the Pell equation x^2 - d*y^2 = +-4 has both x and y even.

The sequence contains the squarefree numbers congruent to 5 modulo 8 that are not in A107997.

This sequence union A107997 = A039955.

This sequence contains all numbers of the form 4*k^2+1 (k > 1) that are squarefree.

Examples

			33 is in the sequence since the fundamental unit of the field Q(sqrt(33)) is 23+4*sqrt(33).
53 is not in the sequence since the fundamental unit of the field Q(sqrt(53)) is 3+omega, where omega = (1+sqrt(53))/2.
		

References

• Z. I. Borevich and I. R. Shafarevich. Number Theory. Academic Press. 1966.

Links

• Keith Matthews, Finding the fundamental unit of a real quadratic field

Crossrefs

Cf. A107997, A039955, A107998, A197170.