Search: a033315 -id:a033315
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A002350
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Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = x, or 1 if n is a square. A002349 gives values of y.
(Formerly M2240 N0890)
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+10
28
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1, 3, 2, 1, 9, 5, 8, 3, 1, 19, 10, 7, 649, 15, 4, 1, 33, 17, 170, 9, 55, 197, 24, 5, 1, 51, 26, 127, 9801, 11, 1520, 17, 23, 35, 6, 1, 73, 37, 25, 19, 2049, 13, 3482, 199, 161, 24335, 48, 7, 1, 99, 50, 649, 66249, 485, 89, 15, 151, 19603, 530, 31, 1766319049, 63, 8, 1
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OFFSET
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1,2
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COMMENTS
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a(p*q^2) = b(p,q/gcd(A002349(p),q)) where
b(p,0) = 1, b(p,1) = a(p), b(p,i) = 2*a(p)*b(p,i-1) - b(p,i-2) for i>1. (End)
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REFERENCES
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A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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EXAMPLE
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For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 0), (9, 4).
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MATHEMATICA
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PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[1]], 1]; Table[ f[n], {n, 0, 65}]
Table[If[! IntegerQ[Sqrt[k]], {k, FindInstance[x^2 - k*y^2 == 1 && x > 0 && y > 0, {x, y}, Integers]}, Nothing], {k, 2, 80}][[All, 2, 1, 1, 2]] (* Horst H. Manninger, Mar 23 2021 *)
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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STATUS
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approved
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A002349
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Take solution to Pellian equation x^2 - n*y^2 = 1 with smallest positive y and x >= 0; sequence gives a(n) = y, or 0 if n is a square. A002350 gives values of x.
(Formerly M0046 N0015)
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+10
24
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0, 2, 1, 0, 4, 2, 3, 1, 0, 6, 3, 2, 180, 4, 1, 0, 8, 4, 39, 2, 12, 42, 5, 1, 0, 10, 5, 24, 1820, 2, 273, 3, 4, 6, 1, 0, 12, 6, 4, 3, 320, 2, 531, 30, 24, 3588, 7, 1, 0, 14, 7, 90, 9100, 66, 12, 2, 20, 2574, 69, 4, 226153980, 8, 1, 0, 16, 8, 5967, 4, 936, 30, 413, 2, 267000, 430, 3
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OFFSET
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1,2
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REFERENCES
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Albert H. Beiler, "The Pellian" (chap 22), Recreations in the Theory of Numbers, 2nd ed. NY: Dover, 1966.
A. Cayley, Report of a committee appointed for the purpose of carrying on the tables connected with the Pellian equation ..., Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 430-443.
C. F. Degen, Canon Pellianus. Hafniae, Copenhagen, 1817.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 55.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
E. E. Whitford, The Pell Equation.
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LINKS
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EXAMPLE
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For n = 1, 2, 3, 4, 5 solutions are (x,y) = (1, 0), (3, 2), (2, 1), (1, 0), (9, 4).
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MATHEMATICA
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a[n_] := If[IntegerQ[Sqrt[n]], 0, For[y=1, !IntegerQ[Sqrt[n*y^2+1]], y++, Null]; y]
(* Second program: *)
PellSolve[(m_Integer)?Positive] := Module[{cof, n, s}, cof = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cof]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[2]], 0]; Table[ f[n], {n, 0, 75}]
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CROSSREFS
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KEYWORD
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nonn,nice,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A033316
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Value of D for incrementally largest values of minimal x satisfying Pell equation x^2-Dy^2=1.
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+10
18
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1, 2, 5, 10, 13, 29, 46, 53, 61, 109, 181, 277, 397, 409, 421, 541, 661, 1021, 1069, 1381, 1549, 1621, 2389, 3061, 3469, 4621, 4789, 4909, 5581, 6301, 6829, 8269, 8941, 9949, 12541, 13381, 16069, 17341, 24049, 24229, 25309, 29269, 30781, 32341, 36061
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OFFSET
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1,2
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COMMENTS
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Equally, value of D for incrementally largest values of minimal y satisfying Pell equation x^2-Dy^2=1.
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LINKS
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MATHEMATICA
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PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; f[n_] := If[ !IntegerQ[ Sqrt[n]], PellSolve[n][[1]], 1]; a = b = -1; t = {}; Do[b = f[n]; If[b > a, t = Append[t, n]; a = b], {n, 1, 40500}]; t
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A336790
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Values of odd prime numbers, D, for incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = -2.
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+10
3
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3, 11, 19, 43, 67, 139, 211, 331, 379, 571, 739, 859, 1051, 1291, 1531, 1579, 1699, 2011, 2731, 3019, 3259, 3691, 3931, 5419, 5659, 5779, 6211, 6379, 6451, 8779, 9619, 10651, 16699, 17851, 18379, 21739, 25939, 32971, 42331, 42571, 44851, 50131, 53299, 55819, 56611, 60811, 61051, 73459, 76651, 90619, 90931
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OFFSET
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1,1
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COMMENTS
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Analogous to A033316 for x^2-D*y^2=1, and D is required to be prime, and for record values of x.
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LINKS
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EXAMPLE
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For D=43, the least x for which x^2-D*y^2=-2 has a solution is 59. The next prime, D, for which x^2-D*y^2=-2 has a solution is 59, but the smallest x in this case is 23, which is less than 59. The next prime, D, after 59 for which x^2-D*y^2=-2 has a solution is 67 and the least x for which it has a solution is 221, which is larger than 59, so it is a new record value. 67 is a term of this sequence and 221 is a term of A336791, but 59 is not a term here because the least x for which x^2-47*y^2=-2 has a solution at D=59 is not a record value.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A336793
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Incrementally largest values of minimal positive y satisfying the equation x^2 - D*y^2 = -2, where D is an odd prime number.
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+10
3
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1, 3, 9, 27, 747, 36321, 2900979, 5843427, 563210019, 11516632737, 48957047673, 953426773899, 23440805582361, 27491112569139, 734940417828177, 1270701455204457, 106719437154440984241, 292398373544007804918339, 62392836359922644036329593, 607918712560763608313068257
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OFFSET
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1,2
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COMMENTS
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For the corresponding numbers D see A336792.
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LINKS
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EXAMPLE
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For D=3, the least positive y for which x^2-D*y^2=-2 has a solution is 1. The next prime, D, for which x^2-D*y^2=-2 has a solution is 11, but the smallest positive y in this case is also 1, which is equal to the previous record y. So 11 is not a term.
The next prime, D, after 11 for which x^2-D*y^2=-2 has a solution is 19 and the least positive y for which it has a solution is y=3, which is larger than 1, so it is a new record y value. So 19 is a term of A336792 and 3 is a term of this sequence.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A336787
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Incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = 2, where D is a prime number.
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+10
2
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2, 3, 5, 39, 59, 477, 2175, 41571, 127539, 340551, 15732537, 221272626669, 2700614460969, 66944775830061, 616049024759241, 6245844517335369, 13085071811371140879, 43795350588094552821, 63464174140920940599, 633160367499665048108061
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OFFSET
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1,1
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COMMENTS
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Analogous to A033315 for x^2 - D*y^2 = 1, and D required to be prime. The x values incrementally largest for x^2 - D*y^2 = 2. D values appear in sequence A336786.
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LINKS
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EXAMPLE
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For D=31, the least x for which x^2 - Dy^2 = 2 has a solution is 39. The next prime, D, for which x^2 - Dy^2 = 2 has a solution is 47, but the smallest x in this case is 7, which is less than 39. The next prime, D, after 47 for which x^2 - Dy^2 = 2 has a solution is 71 and the least x for which it has a solution is x=59, which is larger than 39, a new record value, so 71 is a term of A336786 and 59 is the corresponding term of this sequence. 47 is not a term of A336786 because the least x for which x^2 - 47*y^2 = 2 has a solution is not a record value.
Primes D for which the equation x^2 - D*y^2 = 2 has integer solutions begin 2, 7, 23, 31, 47, 71, 79, 103, ...; at those values of D, the minimal x values satisfying the equation x^2 - D*y^2 = 2 begin as follows:
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x values satisfying minimal
D x^2 - D*y^2 = 2 x value record
--- --------------------------- ------- ------
2 2, 10, 58, 338, 1970, ... 2 *
7 3, 45, 717, 11427, ... 3 *
23 5, 235, 11275, 540965, ... 5 *
31 39, 118521, 360303801, ... 39 *
47 7, 665, 63833, 6127303, ... 7
71 59, 410581, 2857643701, ... 59 *
79 9, 1431, 228951, ... 9
103 477, 217061235, ... 477 *
...
The record high minimal values of x (marked with asterisks) are the terms of this sequence. The corresponding values of D are the terms of A336786. (End)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A336791
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Incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = -2, where D is an odd prime number.
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+10
2
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1, 3, 13, 59, 221, 8807, 527593, 52778687, 113759383, 13458244873, 313074529583, 1434867510253, 30909266676193, 842239594152347, 1075672117707143, 29204057639975683, 52376951398984393, 4785745078256208692917, 15280437983663153103594943
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OFFSET
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1,2
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COMMENTS
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Analogous to A033315 for x^2-D*y^2=1, and D required to be prime.
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LINKS
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EXAMPLE
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For D=43, the least x for which x^2-D*y^2=-2 has a solution is 59. The next prime, D, for which x^2-D*y^2=-2 has a solution is 59, but the smallest x in this case is 23, which is less than 59. The next prime, D, after 59 for which x^2-D*y^2=-2 has a solution is 67 and the least x for which it has a solution is 221, which is larger than 59, so it is a new record value. 67 is a term of A336790 and 221 is a term of this sequence, but 59 is not a term of A336790 because the least x for which x^2-47*y^2=-2 has a solution at D=59 is not a record value.
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MATHEMATICA
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records[n_]:=Module[{ri=n, m=0, rcs={}, len}, len=Length[ri]; While[ len>0, If[ First[ri]>m, m=First[ri]; AppendTo[rcs, m]]; ri=Rest[ri]; len--]; rcs]; records[ Abs[Flatten[Table[x/.FindInstance[x^2-p y^2==-2, {x, y}, Integers], {p, Prime[Range[2, 500]]}]/.x->Nothing]]] (* Harvey P. Dale, Jan 02 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A336795
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Incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = 3, where D is a prime number.
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+10
2
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4, 8, 94, 9532, 289580, 3433342, 57427216, 1610590723242832, 422208570755689121370258391432928, 112180929726349239798469275333193570657564148, 8590101469813781580594707823194303692816416722
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OFFSET
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1,1
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COMMENTS
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Analogous to A033315 for x^2 - D*y^2 = 1, and D required to be prime.
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LINKS
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EXAMPLE
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For D=73, the least x for which x^2 - D*y^2 = 3 has a solution is 94. The next prime, D, for which x^2 - D*y^2 = 3 has a solution is 97, but the smallest x in this case is 10, which is less than 94. The next prime, D, after 97 for which x^2 - D*y^2 = 3 has a solution is 109 and the least x for which it has a solution is 9532, which is larger than 94, so it is a new record value. 73 is a term of A336794 and 94 is a term of this sequence, but 97 is not a term of A336794 because the least x for which x^2 - 97*y^2 = 3 has a solution is not a record value.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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A336800
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Incrementally largest values of minimal y satisfying the equation x^2 - D*y^2 = 3, where D is a prime number.
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+10
2
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1, 11, 913, 23111, 221161, 3450467, 78495388880651, 10727569485920362724490720830137, 2027623752997677729366859925491727716361771, 127194478138610620242010764302143341359067289, 264781463133512691674640873276575271478272395041
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OFFSET
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1,2
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LINKS
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EXAMPLE
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For D=13, the least positive y for which x^2-D*y^2=3 has a solution is 1. The next prime, D, for which x^2-D*y^2=3 has a solution is 61, but the smallest positive y in this case is also 1, which is equal to the previous record y. So, 61 is not a term.
The next prime, D, after 13 for which x^2-D*y^2=3 has a solution is 73 and the least positive y for which it has a solution is y=11, which is larger than 1, so it is a new record y value. So, 73 is a term of A336796 and 11 is a term of this sequence.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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A341076
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Incrementally largest values of minimal x satisfying the equation x^2 - D*y^2 = -3, where D is a prime number.
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+10
2
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0, 2, 7, 11, 13, 5639, 11262809, 1538763335, 126460946201, 1276182285427369, 14786648025753749026871, 105410978030726984449289, 1498381179129960066289070257961, 107744062788861651804382809216696729188191, 2525173635632697805707745894621283442852191
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Analogous to A033315 for x^2 - D*y^2 = 1, and D required to be prime.
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LINKS
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EXAMPLE
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For D=13, the least x for which x^2 - D*y^2 = -3 has a solution is 7. The next prime, D, for which x^2 - D*y^2 = -3 has a solution is 19, but the smallest x in this case is 4, which is less than 7. The next prime, D, after 19 for which x^2 - D*y^2 = -3 has a solution is 31 and the least x for which it has a solution is 11, which is larger than 7, so it is a new record value. x=11 is a term of this sequence and the corresponding value D=31 is a term of A336801, but 19 is not a term there because the least x for which x^2 - D*y^2 = -3 has a solution at D=19 is not a record value.
As D runs through the primes, the minimal x values satisfying the equation x^2 - D*y^2 = -3 begin as follows:
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x values satisfying minimal
D x^2 - D*y^2 = -5 x value record
-- ---------------------- ------- ------
2 (none)
3 0, 3, 12, 45, 168, ... 0 *
5 (none)
7 2, 5, 37, 82, 590, ... 2 *
11 (none)
13 7, 137, 9223, ... 7 *
17 (none)
19 4, 61, 1421, ... 4
23 (none)
29 (none)
31 11, 206, 33646, ... 11 *
37 (none)
41 (none)
43 13, 400, 90932, ... 13 *
...
The record high minimal values of x (marked with asterisks) are the terms of this sequence. The corresponding values of D are the terms of A336801.
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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