Paris, May 12, 2010

A prime number is an integer greater than or equal to 2 that has exactly two distinct natural number divisors, 1 and itself. For example, 2, 3, 5, 7, 11,..., 1789, etc. are prime numbers, whereas 9, divisible by 3, is not a prime number.

Numerous arithmetical problems concern prime numbers and most of them still remain unresolved, sometimes even after several centuries. For example, it has been known since Euclid that the sequence of prime numbers is infinite, but it is still not known if an infinity of prime numbers p exists such that p+2 is also a prime number (problem of twin prime numbers). In the same way, it is not known if there exists an infinity of prime numbers, the decimal representation of which does not use the digit 7.

Two researchers from the Institut de Mathématiques de Luminy (CNRS/Université de la Méditerranée) have recently made an important breakthrough regarding a conjecture formulated in 1968 by the Russian mathematician Alexandre Gelfond concerning the sum of digits of prime numbers. In particular, they have demonstrated that, on average, there are as many prime numbers for which the sum of decimal digits is even as prime numbers for which it is odd.

The methods employed to arrive at this result, derived from combinatorial mathematics, the analytical theory of numbers and harmonic analysis, are highly groundbreaking and should pave the way to the resolution of other difficult questions concerning the representation of certain sequences of integers.

Quite apart from their theoretical interest, these questions are directly linked to the construction of sequences of pseudo-random numbers and have important applications in digital simulation and cryptography.

### Bibliography:

### Contacts:

Numerous arithmetical problems concern prime numbers and most of them still remain unresolved, sometimes even after several centuries. For example, it has been known since Euclid that the sequence of prime numbers is infinite, but it is still not known if an infinity of prime numbers p exists such that p+2 is also a prime number (problem of twin prime numbers). In the same way, it is not known if there exists an infinity of prime numbers, the decimal representation of which does not use the digit 7.

Two researchers from the Institut de Mathématiques de Luminy (CNRS/Université de la Méditerranée) have recently made an important breakthrough regarding a conjecture formulated in 1968 by the Russian mathematician Alexandre Gelfond concerning the sum of digits of prime numbers. In particular, they have demonstrated that, on average, there are as many prime numbers for which the sum of decimal digits is even as prime numbers for which it is odd.

The methods employed to arrive at this result, derived from combinatorial mathematics, the analytical theory of numbers and harmonic analysis, are highly groundbreaking and should pave the way to the resolution of other difficult questions concerning the representation of certain sequences of integers.

Quite apart from their theoretical interest, these questions are directly linked to the construction of sequences of pseudo-random numbers and have important applications in digital simulation and cryptography.

Sur un problème de Gelfond : la somme des chiffres des nombres premiers, (On a Gelfond problem: the sum of digits of prime numbers) C. Mauduit, J. Rivat, Annals of Mathematics, Vol. 171 (2010), No. 3, 1591–1646, May 2010

View web site

Researchers l

Christian Mauduit l T 04 91 26 96 65 l

mauduit@iml.univ-mrs.fr

Joël Rivat l T 04 91 26 95 78 l rivat@iml.univ-mrs.fr

CNRS Press Office l Elsa Champion l T 01 44 96 43 90 l elsa.champion@cnrs-dir.fr

**Latest press releases**

All disciplines

**9 October 2015**

Designing better catalysts more rapidly**30 September 2015**

New electrode gives micro-supercapacitor macro storage capacity**28 September 2015**

Mimicry helps sheep solve a dilemma**23 September 2015**

A scenario that reconciles the Earth with its origins**23 September 2015**

Biologist Eric Karsenti is awarded the 2015 CNRS Gold Medal**22 September 2015**

Decision-making involves a little known brain region**17 September 2015**

First-ever in vitro human spermatogenesis**9 September 2015**

'Hot Jupiter' exoplanets may have formed very rapidly