For various positive integers k, the sums of kth powers of the first n positive integers, Sk(n) := 1 k +2 k +···+n k , are some of the most popular sums in all of mathematics. In this note we prove a congruence modulo n 3 involving two consecutive sums S2k(n) and S2k+1(n). This congruence allows us to establish an equivalent formulation of Giuga’s conjecture. Moreover, if k is even and n ≥ 5 is a prime such that n − 1 ∤ 2k − 2, then this congruence is satisfied modulo n 4 . This suggests a conjecture about when a prime can be a Wolstenholme prime. We also propose several Giuga-Agoh-like conjectures. Further, we establish two congruences modulo n 3 for two binomial-type sums involving sums of powers S2i(n) with i = 0,1,...,k. Finally, we obtain an extension of a result of Carlitz-von Staudt for odd power sums.

https://cs.uwaterloo.ca/journals/JIS/VOL17/Mestrovic/mes4.pdf

On a Congruence Modulo n 3 Involving Two Consecutive Sums of Powers

In 1950 G. Giuga studied the congruence $\sum_{j=1}^{n-1} j^{n-1} \equiv -1$ (mod $n$) and conjectured that it was only satisfied by prime numbers. In this work we generalize Giuga's ideas considering, for each $k \in \mathbb{N}$, the congruence $\sum_{j=1}^{n-1} j^{k(n-1)} \equiv -1$ (mod $n$). It particular, it is proved that a pair $(n,k)\in \mathbb{N}^2$ (with composite $n$) satisfies the congruence if and only if $n$ is a Giuga Number and $ \lambda(n)/\gcd(\lambda(n),n-1)$ divides $k$. In passing, we establish some new characterizations of Giuga Numbers.

Generalizing Giuga's conjecture

A k -strong Giuga number is a composite integer such that ∑
  = 1
  − 1
 j
 n − 1≡ − 1 (mod n). We consider the congruence ∑
  = 1
  − 1
 j
 k(n − 1)≡ − 1 (mod n) for each k $$ \epsilon $$
 ℕ (thus extending Giuga’s ideas for k = 1). In particular, it is proved that a pair (n, k) with composite n satisfies this congruence if and only if n is a Giuga number and ⋋(n) | k(n − 1). In passing, we establish some new characterizations of Giuga numbers and study some properties of the numbers n satisfying ⋋(n) | k(n − 1).

/pdf/variations-on-giuga-numbers-and-giuga-s-congruence-3glo4v8l30.pdf

Variations on Giuga Numbers and Giuga’s Congruence

Our most recent computations tell us that any counterexample to Giuga’s 1950 primality conjecture must have at least 19,908 decimal digits. Equivalently, any number which is both a Giuga and a Carmichael number must have at least 19,908 decimal digits. This bound has not been achieved through exhaustive testing of all numbers with up to 19,908 decimal digits, but rather through exploitation of the properties of Giuga and Carmichael numbers. This bound improves upon the 1996 bound of Borwein, Borwein, Borwein, and Girgensohn. We present the algorithm used, and discuss technical challenges and challenges to further computation.

/pdf/computation-of-an-improved-lower-bound-to-giuga-s-primality-ujbhrw94q4.pdf

Computation of an Improved Lower Bound to Giuga’s Primality Conjecture

A composite positive integer $n$ is said to be a {\it weak Carmichael number} if 
$$ 
\sum_{\gcd(k,n)=1\atop 1\le k\le n-1}k^{n-1}\equiv \varphi(n) \pmod{n}. \leqno(1) 
$$ It is proved that a composite positive integer $n$ is a weak Carmichael number if and only if $p-1\mid n-1$ for every prime divisor $p$ of $n$. This together with Korselt's criterion yields the fact that every Carmichael number is also a weak Carmichael number. 
In this paper we mainly investigate arithmetic properties of weak Carmichael numbers. Motivated by the investigations of Carmichael numbers in the last hundred years, here we establish several related results, notions, examples and computatinoal searches for weak Carmichael numbers and numbers closely related to weak Carmichael numbers. Furthermore, using the software {\tt Mathematica 8}, we present the table containing all non-prime powers weak Carmichael numbers less than $2\times 10^6$. 
Motivated by heuristic arguments, our computations and some old conjectures and results for Carmichael numbers, we propose several conjectures for weak Carmichael numbers and for some other classes of Carmichael like numbers. 
Finally, we consider weak Carmichael numbers in light of Fermat primality test. We believe that it can be of interest to involve certain particular classes of weak Carmichael numbers in some problems concerning Fermat-like primality tests and the generalized Riemann hypothesis.

Generalizations of Carmichael numbers I

We call a family of primes $P$ normal if it contains no two primes $p,q$ such that $p$ divides $q-1$. In this thesis we study two conjectures and their related variants. Guiga's conjecture is that \sum_{k=1}^{n-1} k^{n-1} \equiv n-1 \mod {n)$ implies $n$ is prime. We study a group of eight variants of this equation and derive necessary and sufficient conditions for which they hold. Lehmer's conjecture is that $\phi(n)|n-1$ if and only if $n$ is prime. This conjecture has been verified for up to 13 prime factors of $n$, and we extend this to 14 prime factors. We also examine the related condition $\phi(n)|n+1$ which is known to have solutions with up to 6 prime factors and extend the search to 7 prime factors. For both of these conjectures the set of prime factors of any counterexample $n$ is a normal family, and we exploit this property in our computations.

https://summit.sfu.ca/system/files/iritems1/7394/b18771816.pdf

Computations on normal families of primes

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Computations on normal families of primes