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Carmichael's conjecture on the Euler function is valid below $ 10\sp {10,000,000}$


Authors: Aaron Schlafly and Stan Wagon
Journal: Math. Comp. 63 (1994), 415-419
MSC: Primary 11A25; Secondary 11A51, 11Y11
DOI: https://doi.org/10.1090/S0025-5718-1994-1226815-3
MathSciNet review: 1226815
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Abstract | References | Similar Articles | Additional Information

Abstract: Carmichael's conjecture states that if $ \phi (x) = n$, then $ \phi (y) = n$ for some $ y \ne x$ ($ \phi $ is Euler's totient function). We show that the conjecture is valid for all x under $ {10^{10,900,000}}$. The main new idea is the application of a prime-certification technique that allows us to very quickly certify the primality of the thousands of large numbers that must divide a counterexample.


References [Enhancements On Off] (What's this?)

  • [1] R. D. Carmichael, On Euler's $ \phi $-function, Bull. Amer. Math. Soc. 13 (1907), 241-243. MR 1558451
  • [2] -, The theory of numbers, Wiley, New York, 1914.
  • [3] -, Note on Euler's $ \phi $-function, Bull. Amer. Math. Soc. 28 (1922), 109-110. MR 1560520
  • [4] P. Erdös, Some remarks on Euler's $ \phi $-function, Acta Math. 4 (1958), 10-19.
  • [5] V. Klee, On a conjecture of Carmichael, Bull. Amer. Math. Soc. 53 (1947), 1183-1186. MR 0022855 (9:269d)
  • [6] P. Masai and A. Vallette, A lower bound for a counterexample to Carmichael's conjecture, Boll. Un. Mat. Ital. (6) 1 (1982), 313-316. MR 663298 (84b:10008)
  • [7] R. Pinch, The pseudoprimes up to $ {10^{13}}$ (to appear).
  • [8] C. Pomerance, J. L. Selfridge, and S. S. Wagstaff, Jr., The pseudoprimes to $ 25 \times {10^9}$, Math. Comp. 35 (1980), 1003-1026. MR 572872 (82g:10030)
  • [9] P. Ribenboim, The book of prime number records, Springer-Verlag, New York, 1988. MR 931080 (89e:11052)
  • [10] S. Wagon, Carmichael's "empirical theorem", Math. Intelligencer 8 (1986), 61-63. MR 832597 (87d:11012)
  • [11] -, Mathematica in action, Freeman, New York, 1990.

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Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1994-1226815-3
Keywords: Euler's function, Carmichael's conjecture, prime certification
Article copyright: © Copyright 1994 American Mathematical Society

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