A215615 -id:A215615 - cp's OEIS frontend

A048954 Wendt determinant of n-th circulant matrix C(n).

1, -3, 28, -375, 3751, 0, 6835648, -1343091375, 364668913756, -210736858987743, 101832157445630503, 0, 487627751563388801409591, -4875797582053878382039400448, 58623274842128064372315087290368, -1562716604740038367719196682456673375

Offset: 1

Eric W. Weisstein

det(C(n)) = 0 for n divisible by 6.
The determinant of the circulant matrix is 0 when 6 divides n because the polynomial (x+1)^(6k) - 1 has roots that are roots of unity. See A086569 for a generalization. - T. D. Noe, Jul 21 2003
E. Lehmer claimed and J. S. Frame proved that 2^n - 1 divides a(n) and the quotient abs(a(n))/(2^n - 1) is a perfect square (Ribenboim 1999, p. 128). - Jonathan Sondow, Aug 17 2012
C(n) is the matrix whose first row is [c_1, ..., c_n] where c_i = binomial(n,i-1), and subsequent rows are obtained by cyclically shifting the previous row one place to the right: see examples and PARI code. - M. F. Hasler, Dec 17 2016

			a(2) = det [ 1 2 ; 2 1 ] = -3.
a(3) = det [ 1 3 3 ; 3 1 3 ; 3 3 1 ] = 28.
a(4) = det [ 1 4 6 4 ; 4 1 4 6 ; 6 4 1 4 ; 4 6 4 1 ] = -375.

P. Ribenboim, "Fermat's Last Theorem for Amateurs", Springer-Verlag, NY, 1999, pp. 126, 136.
P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, NY, 1979, pp. 61-63. MR0551363 (81f:10023).

T. D. Noe, Table of n, a(n) for n=1..50
David W. Boyd, The asymptotic behaviour of the binomial circulant determinant, Journal of Mathematical Analysis and Applications, Volume 86, Issue 1, March 1982, Pages 30-38.
E. Brown and M. Chamberland, Generalizing Gauss's Gem, Amer. Math. Monthly, 119 (No. 7, 2012), 597-601. - _N. J. A. Sloane_, Sep 07 2012
D. Burde and W. A. Moens, The structure of Lie algebras with a derivation satisfying a polynomial identity, arXiv:2009.05434 [math.RA], 2020.
L. Carlitz, A determinant connected with Fermat's last theorem, Proc. Amer. Math. Soc. 10 (1959), 686-690.
L. Carlitz, A determinant connected with Fermat's last theorem, Proc. Amer. Math. Soc. 11 (1960), 730-733.
Joshua Cooper and Zhibin Du, Note on the spectra of Steiner distance hypermatrices, arXiv:2403.02287 [math.CO], 2024. See pp. 2, 4.
Greg Fee and Andrew Granville, The prime factors of Wendt's binomial circulant determinant, Math. Comp. 57 (1991), 839-848.
David Ford and Vijay Jha, On Wendt's Determinant and Sophie Germain's Theorem, Experimental Mathematics, 2 (1993) No. 2, 113-120.
J. S. Frame, Factors of the binomial circulant determinant, Fibonacci Quart., 18 (1980), pp. 9-23.
Charles Helou, On Wendt's Determinant, Math. Comp., 66 (1997) No. 219, 1341-1346.
Charles Helou and Guy Terjanian, Arithmetical properties of wendt's determinant, Journal of Number Theory, Volume 115, Issue 1, November 2005, Pages 45-57.
Emma Lehmer, On a resultant connected with Fermat's last theorem, Bull. Amer. Math. Soc. 41 (1935), 864-867.
Gerard P. Michon, Factorization of Wendt's Determinant (table for n=1 to 114)
Anastasios Simalarides, Upper bounds for the prime divisors of Wendt's determinant, Math. Comp., 71 (2002), 415-427.
Eric Weisstein's World of Mathematics, Circulant matrix
E. Wendt, Arithmetische Studien über den letzten Fermatschen Satz, welcher aussagt, dass die Gleichung a^n=b^n+c^n für n>2 in ganzen Zahlen nicht auflösbar ist, Reimer (Berlin), 1894.

Cf. A052182 (circulant of natural numbers), A066933 (circulant of prime numbers), A086459 (circulant of powers of 2), A086569, A129205, A215615, A215616.

See A096964 for another definition.

a[ n_] := Resultant[ x^n - 1, (1+x)^n - 1, x];

{a(n) = if( n<1, 0, matdet( matrix( n, n, i, j, binomial( n, (j-i)%n ))))}

a(n) = polresultant( x^n - 1, (1+x)^n - 1, x )

a(2*n) = A129205(n)^2 * (1-4^n).
a(n) = 0 if and only if 6 divides n. If d divides n, then a(d) divides a(n). - Michael Somos, Apr 03 2007
a(n) = (-1)^(n-1) * (2^n - 1) * A215615(n)^2. - Jonathan Sondow, Aug 17 2012
a(2*n) = -3 * A215616(n)^3. - Jonathan Sondow, Aug 18 2012

Additional comments from Michael Somos, May 27 2000 and Dec 16 2001

A215616 From Wendt's determinant compute (-A048954(2*n)/3)^(1/3).

Original entry on oeis.org

1, 5, 0, 765, 41261, 0, 1175731456, 804611664045, 0, 4133434158867578125, 36792671310208420147421, 0, 33666995638445382179718361163901, 3930778415673723952392425569428439040, 0, 637350736211692642266912139961455499346709367565

Offset: 1

Jonathan Sondow, Aug 17 2012

nonn

It is known that 3 divides A048954(2*n). It is conjectured that the quotient is a perfect cube.

See A048954 for additional comments, references, links, and cross-references.

Gerard P. Michon, Factorization of Wendt's Determinant(see Remarks and Conjectures).

Cf. A048954, A215615.

w[n_] := Resultant[x^n - 1, (1 + x)^n - 1, x]; Table[(-w[2 n]/3)^(1/3), {n, 19}]

a(n) = (-A048954(2*n)/3)^(1/3).

a(n) = 0 if and only if n is divisible by 3.

A129205 From Wendt's determinant compute (A048954(2*n)/(1-4^n))^(1/2).

Original entry on oeis.org

1, 5, 0, 2295, 453871, 0, 545539395584, 4883188189089105, 0, 14214363393075742724609375, 5968603205606800870499639536231, 0, 41302584753289717847206700750464023881130441

Offset: 1

Michael Somos, Apr 03 2007

nonn

P. Ribenboim, 13 Lectures on Fermat's last theorem, Springer-Verlag, NY, 1979, page 62. MR0551363 (81f:10023)

Cf. A048954, A215615, A215616.

{a(n)= if(n<1, 0, n*=2; sqrtint( matdet( matrix( n, n, i, j, binomial( n, (j-i)%n )))/ (1-2^n)))}

a(n)=0 if and only if n is divisible by 3.

a(n) = A215615(2*n). - Jonathan Sondow, Aug 17 2012

A215656 Solution R of (2u)^2 = R^2 - pS^2, where p is the n-th prime of the form 4k+1.

Original entry on oeis.org

147, 20522387091091, 89544370675021535714607142, 8801866915656397716021519532258687362772409962179980790374047406788427

Offset: 1

Jonathan Sondow, Aug 19 2012

nonn

p = A002144(n), u = A215615(p), and S = A215657(n).
A215615 is computed from Wendt's circulant determinant A048954.
Brown and Chamberland (2012, p. 600) give explicit formulas for u, R, S.

			2*A215615(5) = 2*11 = 22 and 22^2  = 147^2 - 5*65^2, so a(1) = 147.

Ezra Brown and Marc Chamberland, Generalizing Gauss's gem, Amer. Math. Monthly, 119 (Aug. 2012), 597-601.

Cf. A002144, A048954, A215615, A215657.

a(n) = sqrt(4*u^2 + p*S^2) with S = A215657(n), p = A002144(n), u = A215615(p).

A215657 Solution S of (2u)^2 = R^2 - pS^2, where p is the n-th prime of the form 4k+1.

Original entry on oeis.org

65, 5691884464123, 2171769991015128035203320, 1634465653492219202324217583600006782459921190308836446038375668451525

Offset: 1

Jonathan Sondow, Aug 20 2012

nonn

p = A002144(n), u = A215615(p), and R = A215656(n).
A215615 is computed from Wendt's circulant determinant A048954.
Brown and Chamberland (2012, p. 600) give explicit formulas for u, R, S.

			2*A215615(5) = 2*11 = 22 and 22^2  = 147^2 - 5*65^2, so a(1) = 65.

Ezra Brown and Marc Chamberland, Generalizing Gauss's gem, Amer. Math. Monthly, 119 (Aug. 2012), 597-601.

Cf. A002144, A048954, A215615, A215656.

a(n) = sqrt((R^2 - 4*u^2)/p) with R = A215656(n), p = A002144(n), u = A215615(p).

cp's OEIS Frontend

A048954 Wendt determinant of n-th circulant matrix C(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula

Extensions

A215616 From Wendt's determinant compute (-A048954(2*n)/3)^(1/3).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A129205 From Wendt's determinant compute (A048954(2*n)/(1-4^n))^(1/2).

Views

Author

Keywords

References

Crossrefs

Programs

PARI

Formula

A215656 Solution R of (2*u)^2 = R^2 - p*S^2, where p is the n-th prime of the form 4k+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A215657 Solution S of (2*u)^2 = R^2 - p*S^2, where p is the n-th prime of the form 4k+1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A215656 Solution R of (2u)^2 = R^2 - pS^2, where p is the n-th prime of the form 4k+1.

A215657 Solution S of (2u)^2 = R^2 - pS^2, where p is the n-th prime of the form 4k+1.