Lehmer's totient problem

Template:Short description Template:For Template:Unsolved In mathematics, Lehmer's totient problem asks whether there is any composite number Template:Mvar such that Euler's totient function Template:Math divides Template:Math. This is an unsolved problem.

It is known that Template:Math if and only if Template:Mvar is prime. So for every prime number Template:Mvar, we have Template:Math and thus in particular Template:Math divides Template:Math. D. H. Lehmer asked in 1932 whether there exist composite numbers with this property.

History

• Lehmer showed that if any composite solution Template:Mvar exists, it must be odd, square-free, and divisible by at least seven distinct primes (i.e. Template:Math). Such a number must also be a Carmichael number.
• In 1980, Cohen and Hagis proved that, for any solution Template:Mvar to the problem, Template:Math and Template:Math.
• In 1988, Hagis showed that if 3 divides any solution Template:Mvar, then Template:Math and Template:Math. This was subsequently improved by Burcsi, Czirbusz, and Farkas, who showed that if 3 divides any solution Template:Mvar, then Template:Math and Template:Math.
• A 2011 result of Luca and Pomerance states that the number of solutions to the problem less than Template:Mvar is at most Template:Math.
• In 2019 Burek and Żmija proved that any solution Template:Mvar satisfies Template:Math.