Consider pi as a prime number

The final digit conspiracy

If you look at the last digits of prime numbers, one rule quickly becomes clear: These atoms of the world of numbers can only end with the digits 1, 3, 7 or 9. All other final digits identify either even numbers or numbers that are divisible by 5. For a long time, mathematicians assumed that the four possible final digits of the prime numbers occur approximately equally often. This is supported by the pseudo-random behavior of the prime numbers in many respects, and the counting of large sets of prime numbers also suggests this.

End digits of consecutive prime numbers. If one looks at large sets of prime numbers, there are strikingly few identical end digits that follow one another. © Podbregar

Unexpected deviations

But prime numbers would not be prime numbers if they didn't step out of line here as well, as researchers only recently discovered. For their study, Kannan Soundararajan and Robert Lemke Oliver from Stanford University examined how often a certain final prime number follows another. In concrete terms: If the first prime number ends with a 1, then the next following prime number should end in a 1, 3,7, or 9 with the same probability. Because the chance for each of these digits is a quarter - at least that is the common assumption.

What the two mathematicians found, however, was something completely different: If the first prime number has the final digit 1, then the second prime number only has an 18 percent probability of ending in a 1. A 3 or 7, on the other hand, occurs in 30 percent of the cases and a 9 in 22 percent. Tests with the other final digits showed similar results. There could be no question of an equal chance or random distribution here.

"Dislike" so far inexplicable

But that means: Successive prime numbers obviously have a kind of “aversion” to the same final digits. “This is completely surprising - and puzzling,” says Soundararajan. Because according to the theory, prime numbers should not be influenced by their neighbors. But at least with prime numbers in the range of up to several trillion this avoidance of the same final digits seems to exist - even if it becomes weaker with larger numerical values. “The prime numbers obviously hate repeating themselves,” says Lemke Oliver.

But what does it mean? One possibility would be that the last digits of prime numbers do not appear the same often. In fact, there are very thin differences in the frequency of the final digits of the prime numbers: Among the first 5.8 million prime numbers, 3 and 7 occur with a frequency of 25.003 percent, 1 and 9, however, “only” 24.997 percent. “But that can't explain it,” says British mathematician James Grime. Because the sequence of 9 and 1 as the final digits of consecutive prime numbers is, according to the new study, the most common combination.

"It seems to be a fundamental property of prime numbers," says Grime. Because this “aversion” to the same final digits occurs in all types of prime numbers. But why? What is the underlying law? So far, mathematicians have no clear answer to this. "I have no idea how one could formulate the correct guess without speculating," says Lemke Oliver.

This peculiarity of the prime numbers remains puzzling for the time being and cannot be explained mathematically.

Nadja Podbregar
Status: June 15, 2018

June 15, 2018