Quadratic Reciprocity

On the law of quadratic reciprocity

Bulletin of the American Mathematical Society ◽

10.1090/s0002-9904-1935-06086-0 ◽

1935 ◽

Vol 41 (6) ◽

pp. 359-361 ◽

Cited By ~ 1

Author(s):

Albert Whiteman

Keyword(s):

Quadratic Reciprocity ◽

The Law

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A note on Gauss’ first proof of the quadratic reciprocity theorem

Proceedings of the American Mathematical Society ◽

10.1090/s0002-9939-1960-0117215-7 ◽

1960 ◽

Vol 11 (4) ◽

pp. 563-563

Author(s):

L. Carlitz

Keyword(s):

Reciprocity Theorem ◽

Quadratic Reciprocity

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Quadratic Reciprocity in a Finite Group

American Mathematical Monthly ◽

10.1080/00029890.2005.11920190 ◽

2005 ◽

Vol 112 (3) ◽

pp. 251-256 ◽

Cited By ~ 5

Author(s):

William Duke ◽

Kimberly Hopkins

Keyword(s):

Finite Group ◽

Quadratic Reciprocity ◽

A Finite Group

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The law of quadratic reciprocity

Proofs from THE BOOK ◽

10.1007/978-3-662-57265-8_5 ◽

2018 ◽

pp. 27-34

Author(s):

Martin Aigner ◽

Günter M. Ziegler

Keyword(s):

Quadratic Reciprocity ◽

The Law

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Note on a product formula for the Bayad function and a law of quadratic reciprocity

Acta Arithmetica ◽

10.4064/aa149-4-2 ◽

2011 ◽

Vol 149 (4) ◽

pp. 321-336

Author(s):

Heima Hayashi

Keyword(s):

Product Formula ◽

Quadratic Reciprocity

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Eisenstein's Misunderstood Geometric Proof of the Quadratic Reciprocity Theorem

College Mathematics Journal ◽

10.1080/07468342.1994.11973576 ◽

1994 ◽

Vol 25 (1) ◽

pp. 29-34 ◽

Cited By ~ 4

Author(s):

Reinhard C. Laubenbacher ◽

David J. Pengelley

Keyword(s):

Reciprocity Theorem ◽

Geometric Proof ◽

Quadratic Reciprocity

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Gauss Sums, Quadratic Reciprocity, and the Jacobi Symbol

A Pythagorean Introduction to Number Theory - Undergraduate Texts in Mathematics ◽

10.1007/978-3-030-02604-2_7 ◽

2018 ◽

pp. 119-130

Author(s):

Ramin Takloo-Bighash

Keyword(s):

Gauss Sums ◽

Quadratic Reciprocity ◽

Jacobi Symbol

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A Minimalist Proof of the Law of Quadratic Reciprocity

American Mathematical Monthly ◽

10.1080/00029890.2019.1655331 ◽

2019 ◽

Vol 126 (10) ◽

pp. 928-928

Author(s):

Bogdan Veklych

Keyword(s):

Quadratic Reciprocity ◽

The Law

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Quadratic reciprocity

Undergraduate Texts in Mathematics - Elements of Number Theory ◽

10.1007/978-0-387-21735-2_9 ◽

2003 ◽

pp. 158-180

Author(s):

John Stillwell

Keyword(s):

Quadratic Reciprocity

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On the septimic character of 2 and 3

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410007715x ◽

1973 ◽

Vol 74 (3) ◽

pp. 421-433 ◽

Cited By ~ 1

Author(s):

Helen Popova Alderson

Keyword(s):

Quadratic Residue ◽

Quadratic Reciprocity ◽

Complicated Conditions

Let e be an integer greater than 1 and let pbe a prime such that p ≡ 1 (mod e). Criteria for 2 to be a residue of degree e modulo p have been obtained in various forms for e = 2, 3, 4 and 5. Thus, Euler and Lagrange proved that 2 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 8). Gauss showed that 2 is a cubic residue mod p if and only if p is representable as p = l2 + 27m2 with integers l and m, and that 2 is a quartic residue mod p if and only if p is representable as p = 12 + 64m2. Lehmer (5) and Alderson (1) have found similar but more complicated conditions for 2 to be a quintic residue mod p. There are analogous results about 3. For example, it follows from quadratic reciprocity that 3 is a quadratic residue mod p if and only if p ≡ ± 1 (mod 12), and Jacobi(4) showed that 3 is a cubic residue mod p if and only if 4p is representable as l2 + 35m2.

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