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

Author

Eric W. Weisstein

Keywords

• sign
• nice

Comments

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

Examples

			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.
		

References

• 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).

Links

• 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.

Crossrefs

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.

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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

Extensions

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