Mathematicians have taken a step forward in understanding patterns within the primes, numbers divisible only by 1 and themselves. According to the new work, the population of prime numbers contains an infinite collection of arithmetic progressions—number sequences in which each term differs from the preceding one by the same fixed amount.
For example, in the sequence 3, 5, 7, each prime number is 2 more than the preceding one. Another example of such a sequence is 5, 11, 17, 23, 29, in which successive primes differ by 6.
For centuries, mathematicians have wondered how many arithmetic progressions such as these exist among the set of prime numbers and how long the progressions can get. In 1939, the Dutch mathematician Johannes van der Corput proved that there are infinitely many progressions with three terms. Whether longer progressions are infinitely plentiful or limited in number and size had remained a matter of conjecture.
The longest known progressions