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Kronecker Symbol

المؤلف:  Ayoub, R. G.

المصدر:  An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963.

الجزء والصفحة:  ...

23-8-2020

1301

Kronecker Symbol

The Kronecker symbol is an extension of the Jacobi symbol  to all integers. It is variously written as  or  (Cohn 1980; Weiss 1998, p. 236) or  (Dickson 2005). The Kronecker symbol can be computed using the normal rules for the Jacobi symbol

			(1)
			(2)
			(3)

plus additional rules for ,

{-1 for n<0; 1 for n>0, " src="https://mathworld.wolfram.com/images/equations/KroneckerSymbol/NumberedEquation1.gif" style="height:41px; width:154px" />

(4)

and . The definition for  is variously written as

{0 for n even; 1 for n odd, n=+/-1 (mod 8); -1 for n odd, n=+/-3 (mod 8) " src="https://mathworld.wolfram.com/images/equations/KroneckerSymbol/NumberedEquation2.gif" style="height:62px; width:233px" />

(5)

or

{0 for 4|n; 1 for n=1 (mod 8); -1 for n=5 (mod 8); undefined otherwise " src="https://mathworld.wolfram.com/images/equations/KroneckerSymbol/NumberedEquation3.gif" style="height:86px; width:223px" />

(6)

(Cohn 1980). Cohn's form "undefines"  for singly even numbers  and , probably because no other values are needed in applications of the symbol involving the binary quadratic form discriminants  of quadratic fields, where  and  always satisfies .

The Kronecker symbol is implemented in the Wolfram Language as KroneckerSymbol[n, m].

The Kronecker symbol  is a real number theoretic character modulo , and is, in fact, essentially the only type of real primitive character (Ayoub 1963).

The illustration above and table below summarize  for , 2, ... and small .

	OEIS	period
	A109017	24	1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, , 0, 0, 0, , 0, , 0, ...
		0	1, , 1, 1, 0, , 1, , 1, 0, , 1, , , 0, 1, , , , 0, ...
		4	1, 0, , 0, 1, 0, , 0, 1, 0, , 0, 1, 0, , 0, 1, 0, , 0, ...
		3	1, , 0, 1, , 0, 1, , 0, 1, , 0, 1, , 0, 1, , 0, 1, , ...
		8	1, 0, 1, 0, , 0, , 0, 1, 0, 1, 0, , 0, , 0, 1, 0, 1, 0, ...
	A034947		1, 1, , 1, 1, , , 1, 1, 1, , , 1, , , 1, 1, ...
0			1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...
1		1	1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2	A091337	8	1, 0, , 0, , 0, 1, 0, 1, 0, , 0, , 0, 1, ...
3	A091338		1, , 0, 1, , 0, , , 0, 1, 1, 0, 1, 1, 0, ...
4	A000035	2	1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...
5	A080891	5	1, , , 1, 0, 1, , , 1, 0, 1, , , 1, 0, ...
6		24	1, 0, 0, 0, 1, 0, , 0, 0, 0, , 0, , 0, 0, 0, , 0, 1, 0, ...

For values of  corresponding to primitive Dirichlet -series , the period of  equals . For , , ..., the periods of  are 0, 8, 3, 4, 0, 24, 7, 8, 0, 40, 11, 6, ... (OEIS A117888) and for , 2, ... they are 1, 8, 0, 2, 5, 24, 0, 8, 3, 40, 0, 12, ... (OEIS A117889). Here, 0 indicates that the sequence is not periodic.

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REFERENCES:

Ayoub, R. G. An Introduction to the Analytic Theory of Numbers. Providence, RI: Amer. Math. Soc., 1963.

Cohn, H. Advanced Number Theory. New York: Dover, p. 35, 1980.

Dickson, L. E. "Kronecker's Symbol." §48 in Introduction to the Theory of Numbers. New York: Dover, p. 77, 1957.

Sloane, N. J. A. Sequences A000035/M0001, A034947, A080891, A091337, A091338, A109017, A117888, and A117889 in "The On-Line Encyclopedia of Integer Sequences."

Weiss, E. Algebraic Number Theory. New York: Dover, 1998.