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Sierpiński's Conjecture

Sierpiński's conjecture on totients asserts that every integer greater than 1 occurs as a value of the totient valence function, or equivalently as a multiplicity. This conjecture was proved by Ford (1998ab).

A different conjecture of Sierpiński, called hypothesis H_1, asserts that for every integer n>=2, writing 1, 2, ..., n^2 in order in an n×n array leaves at least one prime in every row (Schinzel and Sierpiński 1958, p. 206). Equivalently, for k=1, ..., n,

pi(kn)-pi((k-1)n)>=1,

where pi(x) is the prime counting function. This conjecture remains open (Visser 2026) and implies Legendre's conjecture.

See also

Carmichael's Totient Function Conjecture

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References

Erdős, P. "Some Remarks on Euler's phi-Function." Acta Arith. 4, 10-19, 1958.Ford, K. "The Distribution of Totients." Ramanujan J. 2, 67-151, 1998a.Ford, K. "The Distribution of Totients." Electron. Res. Announc. Amer. Math. Soc. 4, 27-34, 1998b.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 94, 1994.Schinzel, A. "Sur l'equation phi(x)=m." Elem. Math. 11, 75-78, 1956.Schinzel, A. and Sierpiński, W. "Sur certaines hypothèses concernant les nombres premiers." Acta Arith. 4, 185-208, 1958. https://doi.org/10.4064/aa-4-3-185-208.Schlafly, A. and Wagon, S. "Carmichael's Conjecture on the Euler Function Is Valid below 10^(10000000)." Math. Comput. 63, 415-419, 1994.Visser, M. "Sierpiński's Hypothesis H_1." 19 Jul 2026. https://arxiv.org/abs/2512.22413.

Referenced on Wolfram|Alpha

Sierpiński's Conjecture

Cite this as:

Weisstein, Eric W. "Sierpiński's Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SierpinskisConjecture.html

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