Carmichael’s totient conjecture

If you know how many positive numbers up to an integer share no common factor above 1 with it, can that count ever identify the integer uniquely? For 5, the numbers 1,2,3,4 all qualify, giving a count of 4. For 8, the qualifying numbers are 1,3,5,7, again giving 4.

Numbers with no common factor above 1 are called coprime; this count is Euler’s totient. Carmichael’s conjecture says every positive integer has a different positive integer with the same count, including the convention that the count for 1 is 1.[1] [2]

Can the count of numbers sharing no common factor above 1 with an integer ever determine that integer uniquely, or must another integer give the same count? The finite comparison $\varphi(5)=\varphi(8)=4$ illustrates a collision and leaves all other counts undecided. Here $\varphi(n)$ counts integers a from 1 through n with $\gcd(a,n)=1$.

Define the multiplicity $F(n)$ as the number of positive integers m with $\varphi(m)=\varphi(n)$. The conjecture is $F(n)\geq 2$ for every $n\geq 1$. This counts integers with an equal value, not the proportion of integers sharing n’s factorization or a claim that every positive number is a totient value.[1] [2]

The comparison of 5 and 8 verifies one shared value, not every possible value. The 2025 paper proves special cases, and known results also produce values with many matching integers. Neither rules out a different value attained by exactly one integer; excluding that exceptional possibility is the remaining universal task.[1] [2]

The Luca–Pollack paper studies the multiplicity function and typical behavior, while the 2025 paper proves selected cases. Results holding for almost all integers allow exceptions, and results realizing every multiplicity at least 2 do not exclude multiplicity 1. The conjecture requires a different preimage for every attained totient value, without a special-form or density restriction.[1] [2]

Counting discards the identities of the numbers counted: four eligible numbers below 5 differ from the four below 8. The conjecture asks whether this loss of information always prevents a unique inverse. It is a concrete test of what an arithmetic statistic can reveal about the integer that produced it.[1] [2]

The totient has a multiplicative description from the prime factors of its input, while multiplicity groups together different inputs with the same output. This juxtaposes local factor arithmetic with the global sizes of the function’s fibers. Typical multiplicity and universal nonuniqueness ask different questions about that inverse structure; both matter even when individual collisions are easy to compute.[1] [2]