Hans Riesel

Some calculations related to Riemann’s prime number formula

Abstract. The objective of this paper is to study the relation of the complex zeros of the Riemann zeta function to the distribution of prime numbers. This relation arises from a formula of Riemann, which is studied here by extensive machine calculations. To establish the validity of the computations, reasonable upper bounds for the various errors involved are deduced. The analysis makes use of a formula, (32), which seems to be quite new. Only the first 29 pairs of complex zeros p = i ia (a < 100), and the primes in the interval x < 106 are considered. It turns out that these zeros of i(s) lead to an approximation of ir(x), the number of primes

Factors of generalized Fermat numbers

A search for prime factors of the generalized Fermat numbers F n (a,b) = a 2n + b 2n has been carried out for all pairs (a, b) with a, b < 12 and GCD(a, b) = 1. The search limit k on the factors, which all have the form p = k 2 m + 1, was k = 10 9 for m < 100 and k = 3. 10 6 for 101 < m < 1000. Many larger primes of this form have also been tried as factors of F n (a, b). Several thousand new factors were found, which are given in our tables. For the smaller of the numbers, i.e. for n < 15, or, if a. b < 8, for n < 16, the cofactors, after removal of the factors found, were subjected to primality tests, an if composite with n ≤ 11, searched for larger factors by using the ECM, and in some cases the MPQS, PPMPQS, or SNFS. As a result all numbers with n ≤ 7 are now completely factored.

Modern Factorization Methods

In this expository paper the progress in factorization of large integers since the introduction of computers is reported. Thanks to theoretical advances and refinements, as well as to more powerful computers, the practical limit of integers possible to factor has been raised considerably during the past 20 years. The present practical limit is around 1075 if supercomputers are used and if much computer time is available.

Summation of double series using the Euler-Maclaurin sum formula

A certain variation of the Euler-Maclaurin sum formula is used to deduce a corresponding formula, suitable for the summation of finite or infinite double series.

Some soluble cases of the discrete logarithm problem

AbstractLetg be a primitive λ-root modn. Then the powersgt mod,n, fort=1, 2, ..., λ(n) represent a (cyclic) subgroupCλ(n) (of order λ(n)) ofMn, the group of order ϕ(n), representingall primitive residue classes modn. To computet backwards fromgt modn is calledthe discrete logarithm problem inCλ(n), also expressible by

On the metric theory of nearest integer continued fractions

a \equiv g^t \bmod n \Leftrightarrow t \equiv \log _g a\bmod \lambda \left( n \right).

Table errata 2 to “Factors of generalized Fermat numbers”

The purpose of this paper is to point out some cases, in which this problem can be solved by astraight-forward formula only, with no element of guessing or collecting factorizations or the like involved at all, taking timeO(log3n) or less to compute (orO(log2+εn), if fast multiplication of multiple precision integers is used).—One very simple case is given by the modulusn=10s, fors≥4. To give just one instance of this case: Forn=108, if the primitive λ-rootg=317 ≡ 29140163 modn is chosen, anda=gt modn, then