Covering system

In mathematics, a covering system is a special way to describe all the whole numbers. It uses groups of numbers, called residue classes, so that every integer is included. Each residue class is written as aᵢ (mod nᵢ), meaning numbers that share the same remainder when divided by nᵢ.

For example, a residue class like 2 (mod 5) includes numbers such as 2, 7, 12, and so on — all numbers that leave a remainder of 2 when divided by 5. When several of these classes are combined, they can cover every integer without missing any.

Covering systems are important in number theory. They help mathematicians solve problems about how numbers relate to each other and find patterns in numbers.

Examples and definitions

The idea of a covering system was first talked about by Paul Erdős in the early 1930s.

A covering system is a special group of numbers. These numbers can stand for every whole number. For example:

The numbers 0, 1, and 2, when looking at remainders after dividing by 3, cover all whole numbers.
The numbers 1 (when dividing by 2), 2 (when dividing by 4), 4 (when dividing by 8), and 0 (when dividing by 8) also cover all whole numbers.

Mathematicians study these systems to learn how numbers can fit together perfectly.

Mirsky–Newman theorem

The Mirsky–Newman theorem is an important idea in mathematics. It says that a special kind of covering system cannot exist where the parts do not overlap. This idea was guessed to be true in 1950 by Paul Erdős. Later, it was shown to be true by Leon Mirsky and Donald J. Newman. Others, like Harold Davenport and Richard Rado, also found the same answer.

Main article: Herzog–Schönheim conjecture

Primefree sequences

Covering systems help us create special sequences of numbers. These sequences follow the same pattern as Fibonacci numbers, but every number in the sequence is a composite number. This means each number can be divided by numbers other than one and itself.

For example, a sequence discovered by Herbert Wilf starts with the numbers 20615674205555510 and 3794765361567513. In this sequence, the positions where numbers can be divided by a certain prime number form a pattern. These patterns together make a covering system, ensuring that every number in the sequence has at least one prime divisor.

Boundedness of the smallest modulus

Mathematician Paul Erdős asked if we could find special number patterns where the smallest number could be made very large. He and others found patterns where the smallest number was 2, 3, 4, 20, 40, and even 42.

Later, Bob Hough showed that there is a limit to how large that smallest number can be. He used a clever mathematical idea called the Lovász local lemma.

Systems of odd moduli

Mathematicians are still trying to answer an important question: can we create a special kind of number system called a covering system using only odd numbers as the rules? This question was proposed by famous mathematicians Erdős and Selfridge. While we don’t know the answer yet, we do know that if such a system exists and uses certain simple odd numbers, it would need to involve at least 22 different prime numbers.