Five of pi’s best fractions have a probable prime on the bottom or on top

Pi’s continued fraction, [3; 7, 15, 1, 292, 1, 1, …], gives a list of fractions that close in on pi, one per step. Their bottom numbers are called q₁, q₂, q₃, …, so q₁ = 7 (from 22/7) and q₃ = 113 (from 355/113). Each is made from the two before it: multiply the last one by the next term of the continued fraction and add the one before that.

The Rhythm of Numbers paper asked whether infinitely many of these bottom numbers are prime. Up to fraction number 20,000 there were ten, the last at number 6,925. The same question can be asked of the top numbers, p₁, p₂, p₃, …: up to fraction 20,000 there were fifteen prime ones, the last at number 8,750. Teddy Rook carried the search to fraction 50,000, top and bottom numbers, with his own programs and found that the bottom numbers q₂₅₃₃₂, q₂₇₆₅₁, q₃₂₃₉₄ and q₃₇₇₂₁ and the top number p₄₂₀₃₇ pass the tests for primes. To save time he first used a “rhythm sieve”: the rule that builds each bottom number, run with the remainders after division by every prime below ten million, which throws out about 19 candidates in 20 before any huge number is touched. You can hear a small version of it on the Rhythm of Numbers page.

We then rebuilt all five numbers from scratch, from two separate high-precision values of pi, and got the same digits. We also repeated the search ourselves, the bottom numbers to fraction 38,571 and the top numbers to fraction 42,037: these are the only new ones that pass there, and our search turned up the top number at 42,037 on its own. Beyond that, to fraction 50,000, the counts rest on his search. In each of the five fractions, the other number is not prime.