# Leon Mirsky, covering systems, and primefree sequences

New on Wikipedia: Leon Mirsky, a Russian-born English mathematician whose interests shifted over time from number theory to linear algebra to combinatorics.

One of the results Mirsky is known for is the Mirsky–Newman theorem, on covering systems, systems of modular congruences that cover all integers. For instance, every integer \( n \) satisfies at least one of the congruences \( n = 1 \pmod{2}, \) \( n = 2 \pmod{4}, \) \( n = 4 \pmod{8}, \) \( n = 2 \pmod{6}, \) \( n = 4 \pmod{12}, \) and \( n = 0 \pmod{24}. \) The Mirsky–Newman theorem states that, in cases like this example where all the moduli are distinct from each other, there must be some numbers (like in this case the number 2) that are covered twice. Equivalently, the sequence of moduli \( m_i \) must have inverses that sum to greater than one: \( 1/2 + 1/4 + 1/8 + 1/6 + 1/12 + 1/24 = 7/6 \gt 1. \)

And one of the things covering systems are good for is finding primefree sequences, sequences of numbers that look like the Fibonacci numbers — they obey the same recurrence and consecutive numbers are relatively prime — but where, unlike the Fibonacci numbers, every number in the sequence is composite. An example of this is Wilf's sequence 20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536, ... where covering systems modulo small primes can be used to show that every number in this sequence has a small prime factor.