In response to a letter from Goldbach, Euler considered sums of the form where s and t are positive integers.As Euler discovered by a process of extrapolation (from s + f g l 3 ) ,

CONTENTS 1. Introduction 2. Numerical Techniques 3. Experimental Setup and Optimizations 4. Integer Relation Detection Algorithms 5. Applications of the PSLQ Algorithm 6. Experimental Results 7. Conjectures Acknowledgments References Borwein was supported by NSERC and the Shrum Endowment at Simon Fraser University. Girgensohn was supported by a DFG fellowship. Euler expressed certain sums of the form 1 X k=1 1 + 1 2 m + + 1 k m (k + 1) ;n , where m and n are positive integers, in terms of the Riemann zeta function. In [Borwein et al. 1993], Euler's results were extended to a significantly larger class of sums of this type, including sums with alternating signs. This research was facilitated by numerical computations using an algorithm that can determine, with high confidence, whether or not a particular numerical value can be expressed as a rational linear combination of several given constants. The present paper presents the numerical techniques used in these computations and lists many of the experimental results that have been obtained.

Let $a,b,c$ be positive integers and define the so-called triple, double and single Euler sums by $$\zeta(a,b,c) \ := \ \sum_{x=1}^{\infty} \sum_{y=1}^{x-1} \sum_{z=1}^{y-1} {1 \over x^a y^b z^c},$$ $$ \zeta(a,b) \ := \ \sum_{x=1}^\infty \sum_{y=1}^{x-1} {1 \over x^a y^b} \quad $$ and $$ \zeta(a) \ := \ \sum_{x=1}^\infty {1 \over x^a}.$$ Extending earlier work about double sums, we prove that whenever $a+b+c$ is even or less than 10, then $\zeta(a,b,c)$ can be expressed as a rational linear combination of products of double and single Euler sums. The proof involves finding and solving linear equations which relate the different types of sums to each other. We also sketch some applications of these results in Theoretical Physics.

G. Giuga conjectured that if an integer n satisfies n−1 k=1 k n−1 ≡ −1 mod n, then n must be a prime. We survey what is known about this interesting and now fairly old conjecture.Giuga proved that n is a counterexample to his conjecture if and only if each prime divisor p of n satisfies (p − 1) | (n/p − 1) and p | (n/p − 1). Using this characterization, he proved computationally that any counterexample has at least 1,000 digits; equipped with more computing power, E. Bedocchi later raised this bound to 1,700 digits. By improving on their method, we determine that any counterexample has at least 13,800 digits.We also give some new results on the second of the above conditions. This leads, in our opinion, to some interesting questions about what we call Giuga numbers and Giuga sequences.

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