An Oxford mathematics PhD has proven a variant of a well-known open problem from number theory: the Gilbreath conjecture. Norman Gilbreath, now 86, discovered something remarkable when he scribbled a series of numbers on a napkin one summer day in 1958. The then 22-year-old math student from Los Angeles wrote down the first as many prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, … Underneath he wrote down the differences of the successive numbers: 1, 2, 2, 4 , 2, 4, 2, … He again wrote down the differences: 1, 0, 2, 2, 2, 2, … Gilbreath performed this process quite a few more times, until only one number remained. He noticed that all those different rows start with the number 1.
Because the number of primes is infinite, you can theoretically continue endlessly with those difference rows. The napkin was the beginning of what came to be called ‘Gilbreath’s conjecture’: each difference row starts with a 1. Two of Gilbreath’s college friends calculated all the difference rows if you start with the first 63,419 prime numbers. They used the SWAC, one of the first computers, then a prized possession of the University of California. The SWAC confirmed the correctness of the conjecture for the first 63,418 difference rows. Later, thanks to clever tricks with powerful computers, the conjecture could be verified for billions of difference rows. But no matter how many rows you verify, there are always infinitely many left.
:format(jpeg):fill(f8f8f8,true)/s3/static.nrc.nl/images/gn4/data81604550-d37411.png)
Also read: How coincidental is the distribution of prime numbers?
No big jumps
Mathematicians have long believed that Gilbreath’s discovery is not limited to prime numbers. The property that – from a certain point in time – every difference sequence starts with a 1 would apply to a broad class of number sequences. Main condition: the differences between numbers may increase, but not by leaps and bounds.
For certain random generated sequences of numbers, Zachary Chase of Oxford University has now proven that property. Just take thirty random numbers between 1 and 9 – use for instance Google to generate them. Write them down one after the other and note the difference row below. Do this a total of 29 times and you’re left with one number. Most likely soon each difference row will consist of only ones and zeros. The chance that this is not the case is miniscule.
Chase turned this into a precise mathematical theorem and sent his proof to a professional journal in 2020. After receiving responses from peer reviewers who verified the manuscript, posted Chase released a revised version on arXiv last month.
Roughly speaking, the theorem comes down to the fact that for every row of numbers that starts with a 2, followed by a 3, and then further according to an exactly defined random process, it holds that from a certain point in time every difference row starts with a 1. The row of numbers should not be too loud. grow. This requirement is laid down in the random process: each number in the row is equal to the previous one plus a randomly drawn number from a set whose upper limit is determined by means of a formula in which logarithms occur. Unlike exponential growth, which is explosive, logarithmic growth is slow.
The coincidence aspect in Chase’s result is essential. Prime numbers are of course not random numbers. But they do behave ‘randomly’ in a sense. It is possible to generate a sequence of numbers by chance, which exhibits the same law as the sequence of prime numbers – the great mathematician Gauss discovered this law as early as the end of the eighteenth century.
Ben Green, Chase’s promoter, made important contributions to understanding the distances between prime numbers. Yet those gaps between primes still harbor all sorts of secrets, leaving Gilbreath’s conjecture unproven. In an email, Chase writes: “Perhaps the distances are behaving in a strange way, so Gilbreath’s conjecture could still be wrong.”