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2024 | OriginalPaper | Chapter

Fermat Quotients and the Ankeny–Artin–Chowla Conjecture

Authors : Nic Fellini, M. Ram Murty

Published in: Class Groups of Number Fields and Related Topics

Publisher: Springer Nature Singapore

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Abstract

The chapter delves into the Ankeny–Artin–Chowla conjecture, which was derived in 1951 and involves four congruence relations for the class number of real quadratic fields. It fills gaps in previous proofs and extends these results, investigating a conjecture proposed by Ankeny, Artin, and Chowla. The text also explores the relationship between Fermat quotients and harmonic numbers, providing a refinement of the original conjecture and demonstrating its unconditional nature. Additionally, it discusses a generalized version of the Ankeny–Artin–Chowla conjecture and its implications, including a connection to Bernoulli numbers and the density of regular primes. The chapter offers a comprehensive treatment of the subject, including detailed proofs and computational tools, making it a valuable resource for specialists in number theory.

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next chapter On the Mod p Iwasawa Theory for Elliptic Curves

Ankeny, N.C., Artin, E., Chowla, S.: The class-number of real quadratic fields. Proc. Natl. Acad. Sci. 37(8), 524–525 (1951)MathSciNetCrossRef

Kiselev, A.A.: An expression for the number of classes of ideals of real quadratic fields by means of Bernoulli numbers. Doklady Akad. Nauk SSSR (N.S.) 61, 777–779 (1948)

Ankeny, N.C., Artin, E., Chowla, S.: The class-number of real quadratic number field. Ann. Math. 56(3), 479–493 (1952)MathSciNetCrossRef

Carlitz, L.: Note on the class number of real quadratic fields. Proc. Am. Math. Soc. 4(4), 535–537 (1953)MathSciNetCrossRef

Washington, L.: Introduction to cyclotomic fields. In: Graduate Texts in Mathematics, vol. 83. Springer, New York (1982)

van der Poorten, A.J., te Riele, H.J.J., Williams, H.C.: Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000. Math. Comput. 70(235), 1311–1328 (2001)MathSciNetCrossRef

Van Der Poorten, A.J., te Riele, H.J.J., Williams, H.C.: Corrigenda and addition to: “Computer verification of the Ankeny-Artin-Chowla conjecture for all primes less than 100 000 000 000” [Math. Comput. 70(235), 1311–1328 (2001); MR1709160 (2001j:11125)]. Math. Comput. 72(241), 521–523 (2003)

Ankeny, N.C., Chowla, S.: A note on the class number of real quadratic fields. Acta Arith 6, 145–147 (1960)MathSciNetCrossRef

Eisenstein, F.: Eine neue Gattung zahlentheoretischer Funktionen, welche von zwei Elementen ahhängen und durch gewisse lineare Funktional-Gleichungen definirt werden. Berichte Königl. Preuß. Akad 15, 36–42 (1850)

10.

Yang, H., Fu, R.: The exponential diophantine equation \(x^y + y^x =z^2\) via a generalization of the Ankeny-Artin-Chowla conjecture. Int. J. Number Theory 14(5), 1223–1228 (2018)MathSciNetCrossRef

11.

Stephens, A.J., Williams, H.C.: Some computational results on a problem concerning powerful numbers. Math. Comput. 50(182), 619–632 (1988)MathSciNetCrossRef

12.

Yu, J., Yu, J.-K.: A note on a geometric analogue of Ankeny-Artin-Chowla’s conjecture. Contemp. Math. 210, 101–105 (1998)MathSciNetCrossRef

13.

Ram Murty, M.: Exponents of Class Groups of Quadratic Fields. Mehta Research Institute of Mathematics and Mathematical Physics, Allahabad, India (1999)

14.

Ram Murty, M., Esmonde, J.: Problems in Algebraic Number Theory, Volume 190 of Graduate Texts in Mathematics, 2nd edn. Springer, New York (2005)

15.

Mordell, L.J.: On a pellian equation conjecture. Acta Arith. 6, 137–144 (1960)MathSciNetCrossRef

16.

Mordell, L.J.: On a Pellian equation conjecture. II. J. Lond. Math. Soc. 36, 282–288 (1961)MathSciNetCrossRef

17.

Slavutskii, ISh.: Real quadratic fields and the Ankeny-Artin-Chowla conjecture. J. Math. Sci. 122(6), 3673–3678 (2004)MathSciNetCrossRef

Title: Fermat Quotients and the Ankeny–Artin–Chowla Conjecture
Authors: Nic Fellini
M. Ram Murty
Publisher: Springer Nature Singapore
Book: Class Groups of Number Fields and Related Topics
Print ISBN: 978-981-9769-10-0

Electronic ISBN: 978-981-9769-11-7

Copyright Year: 2024
DOI: https://doi.org/10.1007/978-981-97-6911-7_1

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