Abstract: On Euclidean division of a prime number by its rank

by Nikos Lygeros

Abstract: Let us note p n the nth prime number and n its rank. A. Vavoda asked a serie of questions on the set of the remainders of euclidean division of a prime number by its rank, among which the question of its finitude. It is Mr. Balazard who has shown that this set is infinite. In addition A. Vavoda asked whether can be found three (or more)consecutive prime numbers having the same remainder for this division. This question was solved by Mr. Balazard in an affirmative way by giving the following solution: 1181=6·194+17, 1187=6·195+17, 1193=6·196+17. This solution, as we will show it thereafter is the smallest solution of the problem of A. Vavoda. Our goal is to explicitly specify a little more this answer by describing the set of solutions of the problem arising for three prime numbers and to give explicit examples up to seven prime numbers in arithmetic pro­gression.