A REMARK ON PRIMALITY TESTING AND DECIMAL EXPANSIONS

A Remark on Primality Testing and Decimal Expansions

We show that for any fixed base a, a positive proportion of primes become composite after any one of their digits in the base a expansion is altered; the case where a = 2 has already been established by Cohen and Selfridge [‘Not every number is the sum or difference of two prime powers’, Math. Comput. 29 (1975), 79–81] and Sun [‘On integers not of the form ±pa ± qb’, Proc. Amer. Math. Soc. 128 ...

A Remark on Primality Testing and the Binary Expansion

We show that for all sufficiently large integers n, a positive fraction of the primes p between 2 and 2 have the property that p− 2 and p+2 are composite for every 0 ≤ i < n − 1. As a consequence, it is not always possible to test whether a number is prime from its binary expansion without reading all of its digits.

Update on Primality Testing

We discuss recent developments in the field of primality testing since the appearance [10] of our joint paper On primes recognizable in deterministic polynomial time.

Primality testing

On the Randomness of Pi and Other Decimal Expansions

Tests of randomness much more rigorous than the usual frequency-of-digit counts are applied to the decimal expansions of π, e and √ 2, using the Diehard Battery of Tests adapted to base 10 rather than the original base 2. The first 10 digits of π, e and √ 2 seem to pass the Diehard tests very well. But so do the decimal expansions of most rationals k/p with large primes p. Over the entire set o...