We prove that if q 5 is an integer, then every qth power of an integer contains at least 5 nonzero digits in its binary expansion. This is a particular instance of one of a collection of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non-Archimedean logarithms with various local arguments.

For an integer b ≥ 2 a real number α is b-normal if, for all m > 0, every m-long string of digits in the base-b expansion of α appears, in the limit, with frequency b−m. Although almost all reals in [0, 1] are b-normal for every b, it has been rather difficult to exhibit explicit examples. No results whatsoever are known, one way or the other, for the class of “natural” mathematical constants, ...

Abstract Addition chains give a very easy way of computing xn knowing x and n. The fact of having a minimal addition chain for an integer n gives the least number of multiplications needed to compute xn. In this paper, we will present the binary method which is optimal for any integer of Hamming weight 1 or 2. We will show that if n has k digits in its binary expansion and the minimal length of...

The first essay discusses, in nontechnical terms, the paradox implicit in defining a random integer as one without remarkable properties, and the resolution of that paradox at the cost of making randomness a property which most integers have but can’t be proved to have. The second essay briefly reviews the search for randomness in the digit sequences of natural irrational numbers like π and art...

We prove that if q > 5 is an integer, then every q-th power of an integer contains at least 5 nonzero digits in its binary expansion. This is a particular instance of one of a collection of rather more general results, whose proofs follow from a combination of refined lower bounds for linear forms in Archimedean and non-Archimedean logarithms with various local arguments.

There is evidence that the ratio of the length of the 2nd and 4th digit (2D:4D) is negatively related to prenatal and adult concentrations of testosterone. It has also been reported that high levels of testosterone at conception in both fathers and mothers are associated with an increased sex ratio (proportion of males at birth). It follows from these observations that low values of 2D:4D may b...

This paper presents an algorithm that computes directly the n-th decimal digit of π in sub-quadratic time and very low memory. It improves previous results of Simon Plouffe, later refined by Fabrice Bellard. The problem of the n-th digit computation in base 2 had already been successfully treated thanks to the use of appropriate series, but no corresponding formula for the question in base 10 h...

The 2nd to 4th digit ratio (2D:4D) is a sexually dimorphic biometric marker, related to prenatal estrogen and testosterone levels in utero, and determined genetically by the HOX genes. 2D:4D presents a population variation, which seems to be dependent on geographical position or ethnicity, and may reflect differences in prenatal steroid hormone levels among different ethnic groups. In view of i...