A107997 -id:A107997 - cp's OEIS frontend

A107998 Squarefree integers m for which the fundamental unit of Q(sqrt(m)) is of the form u + v*sqrt(m) for integer u, v.

2, 3, 6, 7, 10, 11, 14, 15, 17, 19, 22, 23, 26, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 51, 55, 57, 58, 59, 62, 65, 66, 67, 70, 71, 73, 74, 78, 79, 82, 83, 86, 87, 89, 91, 94, 95, 97, 101, 102, 103, 105, 106, 107, 110, 111, 113, 114, 115, 118, 119, 122, 123, 127

Offset: 1

Steven Finch, Jun 13 2005

nonn

H. C. Williams, Eisenstein's problem and continued fractions, Utilitas Math. 37 (1990) 145-157.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
F. Arndt, Beiträge zur Theorie der quadratischen Formen, Archiv der Mathematik und Physik 15 (1850) 467-478.
A. Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod 8), J. Reine Angew. Math. 53 (1857) 369-371.
Steven R. Finch, Class number theory
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Eric Weisstein's World of Mathematics, Fundamental unit

Cf. A107997.

Select[ Range[2, 127], (fu = NumberFieldFundamentalUnits @ Sqrt[#]; SquareFreeQ[#] && IntegerQ[fu[[1, 2, 1]] ] && IntegerQ[fu[[1, 2, 2]] ]) &] (* Jean-François Alcover, Jun 20 2013 *)

A108160 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

Original entry on oeis.org

37, 101, 141, 197, 269, 349, 373, 381, 389, 485, 557, 573, 677, 701, 709, 757, 781, 813, 829, 877, 885, 901, 933, 973, 997, 1149, 1157, 1173, 1213, 1293, 1301, 1389, 1405, 1605, 1613, 1717, 1757, 1765, 1861, 1885, 1893, 1901, 1909, 1949, 1973, 2069, 2077, 2093

Offset: 1

Steven Finch, Jun 13 2005

nonn

C. F. Gauss, Disquisitiones Arithmeticae, Yale Univ. Press, 1966, section 256 VI, pp. 276-277.

Florian Breuer and James Punch, Quadratic units and cubic fields, arXiv:2507.06579 [math.NT], 2025. See p. 1.
Arthur Cayley, Note sur l'équation x^2 - D*y^2 = +-4, D=5 (mod 8), J. Reine Angew. Math. 53 (1857) 369-371.
Steven R. Finch, Class number theory [Cached copy, with permission of the author]
Noburo Ishii, Pierre Kaplan and Kenneth S. Williams, On Eisenstein's problem, Acta Arith. 54 (1990) 323-345.

Cf. A048941, A048942, A107997, A107999.

More terms from Jinyuan Wang, Sep 08 2021

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.

Original entry on oeis.org

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

Emmanuel Vantieghem, Mar 07 2017

nonn

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.

			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.

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

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

Cf. A107997, A039955, A107998, A197170.

A361381 In continued fraction convergents of sqrt(d), where d=A005117(n) (squarefree numbers), the position of a/b where abs(a^2 - d*b^2) = 1 or 4.

Original entry on oeis.org

2, 4, 1, 2, 1, 4, 2, 1, 6, 2, 6, 4, 1, 1, 2, 8, 4, 4, 2, 1, 2, 2, 3, 2, 10, 12, 4, 2, 1, 4, 6, 7, 6, 3, 4, 1, 2, 10, 2, 6, 8, 7, 5, 2, 4, 4, 1, 2, 1, 10, 2, 5, 8, 4, 16, 4, 11, 1, 2, 12, 2, 9, 6, 5, 2, 6, 9, 6, 10, 10, 4, 1, 2, 12, 10, 3, 6, 4, 14, 9, 4, 18, 4, 4, 2, 1, 2, 3, 20, 10, 4, 5, 8, 10, 10, 18, 2, 22

Offset: 5

Ed Pegg Jr, Mar 09 2023

nonn

The golden ratio is the fundamental unit for sqrt(5), but 1/1 isn't a convergent, so this sequence starts with squarefree number A005117(5)=6.

			A005117(13)=19. 170^2 - 19*39^2 = 1. The 6th convergent of sqrt(19) is 170/39.
A005117(14)=21. 5^2   - 21*1^2 =  4. The 2nd convergent of sqrt(21) is 5/1.
A005117(15)=22. 197^2 - 22*42^2 = 1. The 6th convergent of sqrt(22) is 197/42.
A005117(16)=23. 24^2  - 23*5^2 =  1. The 4th convergent of sqrt(23) is 24/5.
Corresponding fundamental units are 170+39*sqrt(19), 5+sqrt(21), 197+42*sqrt(22) and 24+5*sqrt(23).

Cf. A005117, A107997.

f:= proc(x) local CF, k,v,w;
  uses NumberTheory;
  CF:= ContinuedFraction(sqrt(x));
  for k from 0 do
    v:= Convergent(CF,k);
    w:= abs(numer(v)^2 - x*denom(v)^2);
    if w = 1 or w = 4 then return k+1 fi;
  od
end proc:
count:= 0: R:= NULL:
for i from 6 while count < 100 do if NumberTheory:-IsSquareFree(i) then R:= R, f(i); count:= count+1 fi
od:
R; # Robert Israel, Mar 12 2023

(* store A005117 and A107997 first *) Flatten[Table[sqr = Sqrt[A005117[[n]]];
fun = RootReduce[NumberFieldFundamentalUnits[Sqrt[A005117[[n]]]]][[1]];
forcon = If[MemberQ[A107997, A005117[[n]]], RootReduce[2 fun], fun];
converge = Convergents[ContinuedFraction[N[sqr, 140]]];
Flatten[Position[converge, Abs[forcon[[1]]/(forcon[[2]]/ sqr)]]], {n, 4, 101}]]

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A107998 Squarefree integers m for which the fundamental unit of Q(sqrt(m)) is of the form u + v*sqrt(m) for integer u, v.

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A108160 Squarefree integers m congruent to 5 modulo 8 such that the minimal solution of the Pell equation x^2 - m*y^2 = +-4 has both x and y even.

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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.

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A361381 In continued fraction convergents of sqrt(d), where d=A005117(n) (squarefree numbers), the position of a/b where abs(a^2 - d*b^2) = 1 or 4.

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