Problem 48

Are there infinitely many integers $n,m$ such that $\phi(n)=\sigma(m)$ ?

Number theory

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SolvedProved

An accepted solution exists. The statement is true.

PROVED (LEAN). The site's label is PROVED (LEAN). The answer is yes: Ford, Luca and Pomerance proved in 2010 that $\phi$ and $\sigma$ have infinitely many common values (refereed; the accepted claim page is Ford–Luca–Pomerance 2010), and Garaev's refereed sharpening of the count in 2011 is a second accepted claim page (Garaev 2011). The site's Lean marker refers to a community Lean formalization of the Ford–Luca–Pomerance argument, linked from their claim page; this corpus has not built or audited it.

FLP10: Ford, Kevin and Luca, Florian and Pomerance, Carl, Common values of the arithmetic functions $\phi$ and $\sigma$ . Bull. Lond. Math. Soc. (2010), 478-488.
Ga11: Garaev, Moubariz Z., On the number of common values of arithmetic functions $\phi$ and $\sigma$ below $x$ . Mosc. J. Comb. Number Theory 1 (2011), no. 3, 42-49.
Gu04: Guy, Richard K., Unsolved problems in number theory, third edition, Problem Books in Mathematics, Springer (2004), xviii+437 pp.; B38 "Solutions of $\phi(m)=\sigma(n)$ ", printed p. 144: the question as stated here, answered affirmatively by infinitely many twin primes or infinitely many Mersenne primes, with sporadic solutions such as $\phi(780)=192=\sigma(105)$ . Library home: Guy 2004.

Problem 48

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