### First digit counting compatibility II: twin prime powers

#### Abstract

The first digits of twin primes follow a generalized Benford law with size-dependent exponent and tend to be uniformly distributed, at least over the finite range of twin primes up to 10^m, m=5,...,16. The extension to twin prime powers for a fixed power exponent is considered. Assuming the Hardy-Littlewood conjecture on the asymptotic distribution of twin primes, it is claimed that the first digits of twin prime powers associated to any fixed power exponent converge asymptotically to a generalized Benford law with inverse power exponent. In particular, the sequences of twin prime power first digits presumably converge asymptotically to Benfordâ€™s law as the power exponent goes to infinity. Numerical calculations and the analytical first digit counting compatibility criterion support these conjectured statements.

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