On matrix analogs of Fermat’s little theorem. (English. Russian original) Zbl 1136.11022
Math. Notes 79, No. 6, 783-796 (2006); translation from Mat. Zametki 79, No. 6, 840-855 (2006).
The author proves a conjecture attributed to V. I. Arnold which asserts that \(\text{ Tr}(A^{p^r}) \equiv \text{ Tr}(A^{p^{r-1}}) \pmod{p^r}\), where \(A\) is an integer matrix, \(\text{ Tr}\) stands for the trace, \(p\) is a prime number and \(r\) is a positive integer. The proof uses algebraic number theory and is quite involved. The author observes that an alternative (simpler) proof can be obtained from an earlier result of C. J. Smyth [Am. Math. Mon. 93, 469–471 (1986; Zbl 0602.10006)] which asserts that if \(S_d=\alpha_1^d+\dots+\alpha_m^d,\) where \(\alpha_1,\dots,\alpha_m\) are the roots of a monic integer polynomial, then \(\sum_{d| n} \mu(n/d)S_d \equiv 0 \pmod{n}.\) For integer matrices \(A\), the author obtains the following direct analogue of this congruence \( \sum_{d| n} \mu(n/d)\text{ Tr}(A^d) \equiv 0 \pmod{n},\) which is a more general version of Smyth’s result.
We remark that in both (Russian and English) versions there is a misprint in Theorem 10, where the power \(n\) stands instead of the power \(d\) as above.
We remark that in both (Russian and English) versions there is a misprint in Theorem 10, where the power \(n\) stands instead of the power \(d\) as above.
Reviewer: Artūras Dubickas (Vilnius)
MSC:
| 11C20 | Matrices, determinants in number theory |
| 15B36 | Matrices of integers |
| 11A07 | Congruences; primitive roots; residue systems |
| 11R04 | Algebraic numbers; rings of algebraic integers |
| 11R32 | Galois theory |
Citations:
Zbl 0602.10006References:
| [1] | V. I. Arnold, ”Fermat-Euler dynamic dynamical system and the statistics of the arithmetic of geometric progressions,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 37 (2003), no. 1, 1–18. · Zbl 1038.11006 · doi:10.1023/A:1022915825459 |
| [2] | V. I. Arnold, ”Topology of algebra: the combinatorics of squaring,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 37 (2003), no. 3, 20–35. · Zbl 1040.05015 |
| [3] | V. I. Arnold, Euler Groups and the Arithmetic of Geometric Progressions [in Russian], MCCME, Moscow, 2003. |
| [4] | V. I. Arnold, ”Topology and statistics of formulas of arithmetic,” Uspekhi Mat. Nauk [Russian Math. Surveys], 58 (2003), no. 4, 3–28. |
| [5] | V. I. Arnold, ”Fermat dynamics, the arithmetic of matrices, a finite circle, and a finite Lobachevski plane,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 38 (2004), no. 1, 1–15. · Zbl 1125.11066 · doi:10.1023/B:FAIA.0000024863.06462.68 |
| [6] | V. I. Arnold, ”Euler–Fermat matrix theorem,” Izv. Ross. Akad. Nauk Ser. Mat. [Russian Acad. Sci. Izv. Math.], 68 (2004), no. 6, 61–70. |
| [7] | V. I. Arnold, ”Geometry and dynamics of Galois fields,” Uspekhi Mat. Nauk [Russian Math. Surveys], 59 (2004), no. 6, 23–40. |
| [8] | V. I. Arnold, ”Number-theoretic turbulence in Fermat–Euler arithmetics and large Young diagrams geometry statistics,” J. Math. Fluid Mech., 7 (2005), S4–S50. · Zbl 1134.11344 · doi:10.1007/s00021-004-0130-x |
| [9] | V. I. Arnold, ”Ergodic and arithmetical properties of geometric progressions dynamics,” Moscow Math. J. (2005 (to appear)). |
| [10] | V. I. Arnold, ”On the matricial version of Fermat–Euler congruences,” Japanese J. Math. (2005 (to appear)). · Zbl 1168.05003 |
| [11] | Algebraic Number Theory (J. W. S. Cassels and A. Froehlich, Eds.) Academic Press, London, 1967. |
| [12] | Z. I. Borevich and I. R. Shafarevich, Number Theory [in Russian], Nauka, Moscow, 1985. |
| [13] | S. Lang, Algebraic numbers, Addison-Wesley, Reading Mass., 1964. · Zbl 0211.38501 |
| [14] | J.-P. Serre, Corps locaux, Hermann, Paris, 1962. · Zbl 0137.02601 |
| [15] | O. Zariski and P. Samuel, Commutative Algebra. vol. 1, 2, D. Van Nostrand Co. Inc., Princeton, 1958, 1960. · Zbl 0081.26501 |
| [16] | T. Schönemann, ”Grundzüge einer allgemeinen Theorie der höhern Congruenze, deren Modul eine reelle Primzahl ist,” J. Reine Angew. Math., 31 (1846), 269–325. · ERAM 031.0895cj · doi:10.1515/crll.1846.31.269 |
| [17] | S. J. Smyth, ”A coloring proof of a generalization of Fermat’s Little Theorem,” Amer. Math. Monthly, 93 (1986), no. 6, 469–471. · Zbl 0602.10006 · doi:10.2307/2323475 |
| [18] | T. Szele, ”Une généralisation de la congruence de Fermat,” Matematisk tidsskrift, B (1948), 57–59. · Zbl 0032.01103 |
| [19] | L. E. Dickson, History of the Theory of Numbers. V. 1, Chelsea, New York, 1971. |
| [20] | V. V. Prasolov, Polynomials [in Russian], MCCME, Moscow, 2003. |
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