PUMaC 2021 · 数论（A 组） · 第 2 题

PUMaC 2021 — Number Theory (Division A) — Problem 2

A substring of a number n is a number formed by removing some digits from the beginning and end of n (possibly a different number of digits is removed from each side). Find the sum of all prime numbers p that have the property that any substring of p is also prime.

解析

A substring of a number n is a number formed by removing any number of digits from the beginning and end of n (not necessarily the same number of digits are removed from each side). Find the sum of all prime numbers p that have the property that any substring of p is also prime. Proposed by: Daniel Carter Answer: 576 The prime numbers in question are 2, 3, 5, 7, 23, 37, 53, 73, and 373, which sum to 576. One can find the one- and two-digit primes with this property without much difficulty. Given those, the only candidate three-digit numbers are 237, 373, 537, and 737, of which only 373 is prime. Then one can see immediately that there are no four-digit primes with this property, since both the first and last three digits must also be primes with this property, i.e. they must both be 373. This also means there are no primes with five or more digits with this property. 2 2