A094049 -id:A094049 - cp's OEIS frontend

A094048 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least m for which m^2+1=p(n)*k^2 has a solution.

2, 18, 4, 70, 6, 32, 182, 29718, 1068, 500, 5604, 10, 8890182, 776, 1744, 113582, 4832118, 1118, 1111225770, 1764132, 14, 1710, 23156, 71011068, 16, 82, 8920484118, 1063532, 2482, 126862368, 352618

Offset: 1

Matthijs Coster, Apr 29 2004

nonn

Subsequence of A191860. [Reinhard Zumkeller, Jun 18 2011]

Cf. A002144, A094049 (associated k), A130226, A137351, A179073.

Cf. A000196, A037213, A000290.

a094048 n = head [m | m <- map (a037213 . subtract 1 . (* a002144 n))
                               (tail a000290_list), m > 0]
-- Reinhard Zumkeller, Jun 13 2015

f[n_] := Block[{y = 1}, While[ !IntegerQ[ Sqrt[n*y^2 - 1]], y++ ]; Sqrt[n*y^2 - 1]]; lst = {}; Do[p = Prime@ n; If[ Mod[p, 4] == 1, AppendTo[lst, f@p]; Print[{n, Prime@n, f@p}]], {n, 66}]; lst

Edited by Don Reble, Apr 30 2004

A228077 Determinant of the (p_n-1)/2 X (p_n-1)/2 matrix with (i,j)-entry being the Legendre symbol ((j-i)/p_n), where p_n is the n-th prime.

Original entry on oeis.org

0, -1, 0, 0, -5, 1, 0, 0, -13, 0, -145, 5, 0, 0, -25, 0, -3805, 0, 0, 125, 0, 0, 53, 569, -401, 0, 0, -851525, 73, 0, 0, 149, 0, -9305, 0, -385645, 0, 0, -85, 0, -82596761, 0, 126985, -785, 0, 0, 0, 0, -1321693313, 1517, 0, 4574225, 0, 1025, 0, -134485, 0, -535979945, 63445, 0, -145, 0, 0, 7170685, -19805, 0, 55335641, 0, -167273125693, 3793, 0, 0, -27765559705, 0, 0, -427316305, -1027776565, 2564801, 5534176685

Offset: 2

Zhi-Wei Sun, Aug 09 2013

sign

Conjecture: In the case p_n == 1 (mod 4), (2/p_n)*a(n) is a positive odd integer whose prime factors are all congruent to 1 modulo 4, and moreover for some integer b(n) we have b(n) + (2/p_n)*a(n)*sqrt(p_n) = e(p_n)^{(2-(2/p_n))h(p_n)}, where e(p_n) and h(p_n) are the fundamental unit and the class number of the real quadratic field Q(sqrt(p_n)) respectively.
Note that a(n) = 0 when p_n == 3 (mod 4), this is because the transpose of the determinant a(n) coincides with (-1/p_n)^{(p_n-1)/2}*a(n) = -a(n).
M. Vsemirnov has proved Robin Chapman's conjecture on the evaluation of the determinant of the (p+1)/2-by-(p+1)/2 matrix with (i,j)-entry (i,j = 0,...,(p-1)/2) being the Legendre symbol ((j-i)/p), where p is an odd prime.
On Aug 14 2013, Robin Chapman informed the author that he first made the conjecture about the exact value of a(n) in a private manuscript dated Aug 05 2003.

			a(2) = 0 since the Legendre symbol ((1-1)/3) is zero.

Zhi-Wei Sun, Table of n, a(n) for n = 2..200
R. C. Chapman, My evil determinant problem, preprint, 2012.
Zhi-Wei Sun, On some determinants with Legendre symbol entries, arXiv:1308.2900 [math.NT], 2010-2013; Finite Fields Appl. 56(2019), 285-307.
Maxim Vsemirnov, On the evaluation of R. Chapman's "evil determinant", Linear Algebra Appl. 436(2012), 4101-4106.
Maxim Vsemirnov, On R. Chapman's "evil determinant": case p == 1 (mod 4), Acta Arith. 159(2013), 331-344.

Cf. A226163, A227609, A227968, A227971, A228005.

Cf. A094049.

a[n_]:=Det[Table[JacobiSymbol[j-i,Prime[n]],{i,1,(Prime[n]-1)/2},{j,1,(Prime[n]-1)/2}]]
Table[a[n],{n,2,20}]

a(n) = my(p=prime(n)); matdet(matrix((p-1)/2, (p-1)/2, i, j, i--; j--; kronecker(i-j, p))); \\ Michel Marcus, Aug 25 2021

A306529 x-value of the smallest solution to x^2 - py^2 = 2(-1)^((p+1)/4), p = A002145(n).

Original entry on oeis.org

1, 3, 3, 13, 5, 39, 59, 7, 23, 221, 59, 9, 9, 477, 31, 2175, 103, 8807, 41571, 8005, 13, 2047, 2999, 127539, 527593, 15, 15, 2489, 1917, 373, 340551, 11759, 9409, 4109, 52778687, 801, 19, 137913, 113759383, 137, 16437, 12311, 21, 21, 15732537, 1275, 1729, 7204587, 305987, 67

Offset: 1

Jianing Song, Mar 25 2019

nonn

a(n) exists for all n.

X = a(n)^2 - (-1)^((p+1)/4), Y = a(n)*A306566(n) gives the smallest solution to x^2 - p*y^2 = 1, p = A002145(n). As a result, all the positive solutions to x^2 - p*y^2 = 2*(-1)^((p+1)/4) are given by (x_n, y_n) where x_n + (y_n)*sqrt(p) = (a(n) + A306566(n)*sqrt(p))*(X + Y*sqrt(p))^n.

			The smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |      Equation     | x_min | y_min
  1 | x^2 -  3*y^2 = -2 |     1 |     1
  2 | x^2 -  7*y^2 = +2 |     3 |     1
  3 | x^2 - 11*y^2 = -2 |     3 |     1
  4 | x^2 - 19*y^2 = -2 |    13 |     3
  5 | x^2 - 23*y^2 = +2 |     5 |     1
  6 | x^2 - 31*y^2 = +2 |    39 |     7
  7 | x^2 - 43*y^2 = -2 |    59 |     9
  8 | x^2 - 47*y^2 = +2 |     7 |     1
  9 | x^2 - 59*y^2 = -2 |    23 |     3

Cf. A002145, A306566 (y-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306618, A306619 (2*x^2 - A002145(n)*y^2 = (-1)^((p+1)/4)).

b(p) = if(isprime(p)&&p%4==3, x=1; while(!issquare((x^2 - 2*(-1)^((p+1)/4))/p), x++); x)
forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))

If the continued fraction of sqrt(A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then a(n) = x and A306566(n) = y.

A306566 y-value of the smallest solution to x^2 - py^2 = 2(-1)^((p+1)/4), p = A002145(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 7, 9, 1, 3, 27, 7, 1, 1, 47, 3, 193, 9, 747, 3383, 627, 1, 153, 217, 9041, 36321, 1, 1, 161, 121, 23, 20687, 699, 537, 233, 2900979, 43, 1, 7199, 5843427, 7, 803, 593, 1, 1, 731153, 59, 79, 326471, 13809, 3, 7, 12507, 541137, 11, 563210019

Offset: 1

Jianing Song, Mar 25 2019

nonn

a(n) exists for all n.

X = A306529(n)^2 - (-1)^((p+1)/4), Y = A306529(n)*a(n) gives the smallest solution to x^2 - p*y^2 = 1, p = A002145(n). As a result, all the positive solutions to x^2 - p*y^2 = 2*(-1)^((p+1)/4) are given by (x_n, y_n) where x_n + (y_n)*sqrt(p) = (A306529(n) + a(n)*sqrt(p))*(X + Y*sqrt(p))^n.

			The smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |      Equation     | x_min | y_min
  1 | x^2 -  3*y^2 = -2 |     1 |     1
  2 | x^2 -  7*y^2 = +2 |     3 |     1
  3 | x^2 - 11*y^2 = -2 |     3 |     1
  4 | x^2 - 19*y^2 = -2 |    13 |     3
  5 | x^2 - 23*y^2 = +2 |     5 |     1
  6 | x^2 - 31*y^2 = +2 |    39 |     7
  7 | x^2 - 43*y^2 = -2 |    59 |     9
  8 | x^2 - 47*y^2 = +2 |     7 |     1
  9 | x^2 - 59*y^2 = -2 |    23 |     3

Cf. A002145, A306529 (x-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306618, A306619 (2*x^2 - A002145(n)*y^2 = (-1)^((p+1)/4)).

b(p) = if(isprime(p)&&p%4==3, y=1; while(!issquare(p*y^2 + 2*(-1)^((p+1)/4)), y++); y)
forprime(p=3, 500, if(p%4==3, print1(b(p), ", ")))

If the continued fraction of sqrt(A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then A306529(n) = x and a(n) = y.

A306618 x-value of the smallest solution to 2x^2 - py^2 = (-1)^((p+1)/4), p = A002145(n).

Original entry on oeis.org

1, 2, 7, 3, 78, 4, 51, 732, 277, 191, 6, 44, 20621, 122, 416941, 8, 5123, 25, 1034, 9, 3993882, 210107, 203100, 10, 1325, 5248, 65030839, 20107956, 30953, 4584105462, 1036, 4889, 295081, 58746, 20725, 98465863939, 1494439626, 1612, 10173, 6040149252, 102607, 9460742124

Offset: 1

Jianing Song, Mar 25 2019

nonn

a(n) exists for all n.

X = 4*a(n)^2 - (-1)^((p+1)/4), Y = 2*a(n)*A306619(n) gives the smallest solution to x^2 - 2p*y^2 = 1, p = A002145(n).

			The smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |       Equation      | x_min | y_min
  1 | 2*x^2 -  3*y^2 = -1 |     1 |     1
  2 | 2*x^2 -  7*y^2 = +1 |     2 |     1
  3 | 2*x^2 - 11*y^2 = -1 |     7 |     3
  4 | 2*x^2 - 19*y^2 = -1 |     3 |     1
  5 | 2*x^2 - 23*y^2 = +1 |    78 |    23
  6 | 2*x^2 - 31*y^2 = +1 |     4 |     1
  7 | 2*x^2 - 43*y^2 = -1 |    51 |    11
  8 | 2*x^2 - 47*y^2 = +1 |   732 |   151
  9 | 2*x^2 - 59*y^2 = -1 |   277 |    51

Cf. A002145, A306619 (y-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306529, A306566 (x^2 - A002145(n)*y^2 = 2*(-1)^((p+1)/4)).

b(p) = if(isprime(p)&&p%4==3, x=1; while(!issquare((2*x^2 - (-1)^((p+1)/4))/p), x++); x)
forprime(p=3, 250, if(p%4==3, print1(b(p), ", ")))

If the continued fraction of sqrt(2*A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then a(n) = x/2 and A306619(n) = y.

A306619 y-value of the smallest solution to 2x^2 - py^2 = (-1)^((p+1)/4), p = A002145(n).

Original entry on oeis.org

1, 1, 3, 1, 23, 1, 11, 151, 51, 33, 1, 7, 3201, 17, 57003, 1, 633, 3, 119, 1, 437071, 22209, 20783, 1, 129, 497, 6104097, 1839433, 399752993, 89, 411, 23817, 4711, 1611, 7475426163, 111543983, 119, 739, 436478927, 7089, 644468311, 103, 93487270491, 573497, 57, 4182991

Offset: 1

Jianing Song, Mar 25 2019

nonn

a(n) exists for all n.

X = 4*A306618(n)^2 - (-1)^((p+1)/4), Y = 2*A306618(n)*a(n) gives the smallest solution to x^2 - 2p*y^2 = 1, p = A002145(n).

			The smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4) for the first primes congruent to 3 modulo 4:
  n |       Equation      | x_min | y_min
  1 | 2*x^2 -  3*y^2 = -1 |     1 |     1
  2 | 2*x^2 -  7*y^2 = +1 |     2 |     1
  3 | 2*x^2 - 11*y^2 = -1 |     7 |     3
  4 | 2*x^2 - 19*y^2 = -1 |     3 |     1
  5 | 2*x^2 - 23*y^2 = +1 |    78 |    23
  6 | 2*x^2 - 31*y^2 = +1 |     4 |     1
  7 | 2*x^2 - 43*y^2 = -1 |    51 |    11
  8 | 2*x^2 - 47*y^2 = +1 |   732 |   151
  9 | 2*x^2 - 59*y^2 = -1 |   277 |    51

Cf. A002145, A306618 (x-values).

Similar sequences: A094048, A094049 (x^2 - A002144(n)*y^2 = -1); A306529, A306566 (x^2 - A002145(n)*y^2 = 2*(-1)^((p+1)/4)).

b(p) = if(isprime(p)&&p%4==3, y=1; while(!issquare((p*y^2 + (-1)^((p+1)/4))/2), y++); y)
forprime(p=3, 250, if(p%4==3, print1(b(p), ", ")))

If the continued fraction of sqrt(2*A002145(n)) is [a_0; {a_1, a_2, ..., a_(k-1), a_k, a_(k-1), ..., a_1, 2*a_0}], where {} is the periodic part, let x/y = [a_0; a_1, a_2, ..., a_(k-1)], gcd(x, y) = 1, then A306618(n) = x/2 and a(n) = y.

cp's OEIS Frontend

A094048 Let p(n) be the n-th prime congruent to 1 mod 4. Then a(n) = the least m for which m^2+1=p(n)*k^2 has a solution.

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A228077 Determinant of the (p_n-1)/2 X (p_n-1)/2 matrix with (i,j)-entry being the Legendre symbol ((j-i)/p_n), where p_n is the n-th prime.

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A306529 x-value of the smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4), p = A002145(n).

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A306566 y-value of the smallest solution to x^2 - p*y^2 = 2*(-1)^((p+1)/4), p = A002145(n).

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A306618 x-value of the smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4), p = A002145(n).

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A306619 y-value of the smallest solution to 2*x^2 - p*y^2 = (-1)^((p+1)/4), p = A002145(n).

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A306529 x-value of the smallest solution to x^2 - py^2 = 2(-1)^((p+1)/4), p = A002145(n).

A306566 y-value of the smallest solution to x^2 - py^2 = 2(-1)^((p+1)/4), p = A002145(n).

A306618 x-value of the smallest solution to 2x^2 - py^2 = (-1)^((p+1)/4), p = A002145(n).

A306619 y-value of the smallest solution to 2x^2 - py^2 = (-1)^((p+1)/4), p = A002145(n).