Progressive Primes

Prime numbers have all sorts of remarkable and mysterious properties.

Evenly divisible only by themselves and 1, primes can’t be written as the product of smaller positive integers. There are infinitely many of them, and they appear to be scattered somewhat haphazardly among the whole numbers.

It’s not yet known if there are infinitely many twin primes—pairs of primes that are only 2 apart. Or whether every even integer greater than 2 can be written as the sum of two primes. Or whether there’s always a prime between n2 and (n + 1)2. But it is known that there’s always a prime between n and 2n.

In one step toward elucidating certain primal mysteries, two mathematicians have now apparently proved that the population of primes contains an infinite collection of arithmetic progressions.

An arithmetic progression is a sequence of numbers in which each term differs from the preceding one by the same fixed amount. For example, 1, 5, 9, 13, 17, and 21 is an arithmetic sequence in which consecutive numbers differ by 4.

A prime arithmetic progression is one in which the numbers are all primes. For example, 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089 is a 10-term arithmetic progression in which primes differ by 210.

In 1939, Johannes G. van der Corput (1890–1973) proved that there are infinitely many prime arithmetic progressions consisting of precisely three terms, such as 3, 5, 7 or 7, 13, 19 or 61, 67, 73. The new result greatly expands this realm, showing that the primes contain arithmetic progressions of any given length.

However, the proffered proof, by Ben Green of the Pacific Institute of the Mathematical Sciences in Vancouver and Terence Tao of the University of California, Los Angeles, establishes only that such sequences exist. It doesn’t specify how to find these sequences or where they lie among the primes.

The longest known prime arithmetic progressions have just 22 terms. One was found in 1993. It starts with the prime 11,410,337,850,553, and successive primes are 4,609,098,694,200 apart. A second sequence, discovered in 2003, starts with 376,859,931,192,959, and successive terms are 18,549,279,769,020 apart.

Someone may yet come up with a general-purpose method for identifying longer prime arithmetic progressions. Until that happens, however, it’s still nice to know that these sequences exist.

You can also look for sequences of consecutive primes in arithmetic progression. For example, the sequence 251, 257, 263, 269 consists of four consecutive primes, with successive terms differing by 6. Finding such sequences of consecutive primes presents a much tougher challenge than that of finding prime arithmetic progressions.

The current record, set in 1998, consists of 10 primes in a row. The record-holding sequence starts with the 93-digit number 100,996,972,469,714,247,637,786,655,587,969,840,329,509,324,
689,190,041,803,603,417,758,904,341,703,348,882,159,067,229,719, with successive primes differing by 210.

No one yet knows if there’s an arithmetic progression of consecutive primes for any number of terms. Or even whether there are infinitely many sets of three consecutive primes in arithmetic progression.

There seems to be no end to the questions that can be asked of prime numbers. Indeed, as Richard K. Guy of the University of Calgary once remarked, “With most good mathematical problems, the more you solve, the more new problems are propagated.”

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