# First digit counting compatibility II: twin prime powers

• Werner Hurlimann Swiss Mathematical Society, CH-1700 Fribourg, Switzerland
Keywords: first digit, twin primes, Hardy-Littlewood conjecture, probabilistic number theory, asymptotic distribution, mean absolute deviation, probability weighted least squares

### 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 Benford’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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