Joint IAS/Princeton University Number Theory Seminar

Write $a \prec b$ if $b \equiv 1 \pmod{a}$. A prime chain is a chain $p_1 \prec p_2 \prec \dotsb \prec p_k$ whose components are prime numbers. In my talk, I will sketch the proof of the fact that the number of prime chains above $p$ (i.e., with $p_1 = p$ and having $p_k \leq x$) is $\ll_\epsilon (x/p)^{1+\epsilon}$. This is joint work with Ford and Konyagin. I shall also indicate how this estimate entered the proofs of two recent results in elementary number theory, one of them being that there are infinitely many common values of the Euler function $\phi(a)$ and the sum of divisor function $\sigma(b)$ (joint with Ford and Pomerance), and the other being a “near proof” of the validity of the Carmichael conjecture for the Carmichael function $\lambda(n)$ to the effect that for all $n$ there is $m \neq n$ with $\lambda(m) = \lambda(n)$ (joint with Ford).