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It is well-known that the congruence ∑ i = 1 n i n ≡ 1 ( mod n ) has exactly five solutions: { 1 , 2 , 6 , 42 , 1806 } . In this work, we characterize the solutions to the congruence 1 n + 2 n + ⋯ + n n ≡ p ( mod n ) for every prime p . This characterization leads to an algorithm for computing all such solutions, when there is a finite number of them. More generally, our algorithm enables computing all the solutions below a much higher bound as compared to what can be achieved by a naive exhaustive search.

There are n independent Bernoulli random variables with parameters $$p_i$$ that are observed sequentially. We consider the following sequential two-person zero-sum game. Two players, A and B, act in turns starting with player A. The game has n stages, at stage k, if $$ I_k = 1 $$ , then the player having the turn can choose either to keep the turn or to pass it to the other player. If the $$ I_k = 0 $$ , then the player with the turn is forced to keep it. The aim of the game is not to have the turn after the last stage: that is, the player having the turn at stage n wins if $$I_n=1$$ and, otherwise, he loses. We determine the optimal strategy for the player whose turn it is and establish the necessary and sufficient condition for player A to have a greater probability of winning than player B. We find that, in the case of n Bernoulli random variables with parameters 1 / n, the probability of player A winning is decreasing with n toward its limit $$\frac{1}{2} -\frac{1}{2\,e^2}=0.4323323\ldots $$ . We also study the game when the parameters are the results of uniform random variables, $$\mathbf {U}[0,1]$$ .

We consider a version of the secretary problem in which each candidate has an identical twin. As in the classical secretary problem, the aim is to choose a top candidate, i.e., one of the best twins, with the highest possible probability. We find an optimal stopping rule for such a choice, the probability of success, and its asymptotic behavior. We use a novel technique that allows the problem to be solved exactly in linear time and obtain the asymptotic values by solving differential equations. Furthermore, the proposed technique may be used to study the variants of the same problem and in other optimal stopping problems.

In this paper, we deal with the problem of computing the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] with [Formula: see text] and [Formula: see text] a finite commutative ring. We completely solve the problem in the case [Formula: see text] and give some results that compute the value of this sum if [Formula: see text] is an arbitrary finite commutative ring for many values of [Formula: see text] and [Formula: see text]. Finally, based on computational evidence and using some technical results proved in this paper, we conjecture that the sum of the [Formula: see text]th powers of all the elements of the matrix ring [Formula: see text] is always [Formula: see text] unless [Formula: see text], [Formula: see text], [Formula: see text] and the only element [Formula: see text] such that [Formula: see text] is idempotent, in which case the sum is [Formula: see text].

This article studies the zero divisor graphs of the ring of Lipschitz integers modulo n. In particular we focus on the number of vertices, the diameter and the girth. We also give some results regarding the domination number of these graphs.

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).

In this paper we investigate the structure of \({\left(\frac{a,b}{\mathbb{Z}/n \mathbb{Z}}\right)}\), the quaternion rings over \({\mathbb{Z}/n \mathbb{Z}}\). It is proved that these rings are isomorphic to \({\left(\frac{-1,-1}{\mathbb{Z}/n \mathbb{Z}}\right)}\) if \({ a \equiv b \equiv -1 (\mod {4})}\) or to \({\left(\frac{1,1}{\mathbb{Z}/n \mathbb{Z}}\right)}\) otherwise. We also prove that the ring \({\left(\frac{a,b}{\mathbb{Z}/n \mathbb{Z}}\right)}\) is isomorphic to \({\mathbb{M}_2(\mathbb{Z}/n \mathbb{Z})}\) if and only if n is odd and that all quaternion algebras defined over \({\mathbb{Z}/n \mathbb{Z}}\) are isomorphic if and only if \({n \not \equiv 0 (\mod {4})}\) .

We consider a generalization of the quaternion ring $$\Big (\frac{a,b}{R}\Big )$$ over a commutative unital ring R that includes the case when a and b are not units of R. In this paper, we focus on the case $$R=\mathbb {Z}/n\mathbb {Z}$$ for and odd n. In particular, for every odd integer n we compute the number of non R-isomorphic generalized quaternion rings $$\Big (\frac{a,b}{\mathbb {Z}/n\mathbb {Z}}\Big )$$ .

In this paper we study the sum of powers of the Gaussian integers \(\mathbf {G}_k(n):=\sum _{a,b \in [1,n]} (a+b i)^k\). We give an explicit formula for \(\mathbf {G}_k(n) \pmod n\) in terms of the prime numbers \(p \equiv 3 \pmod 4\) with \(p \mid \mid n\) and \(p-1 \mid k\), similar to the well known one due to von Staudt for \(\sum _{i=1}^n i^k \pmod n\). We apply this result to study the set of integers \(n\) which divide \(\mathbf {G}_n(n)\) and compute its asymptotic density with six exact digits: \(0.971000\ldots \).

There are n independent Bernoulli random variables I k with parameters p k that are observed sequentially. We consider an extension of the last-success-problem with reward w k if the player predicts correctly at step k that I k = 1 is the last success. We establish the optimal strategy for a payoff-function generalizing the last-success 0 − 1 payoff by using the dynamical programming method. In particular we show that this method is intuitive and very efficient for general payoffs.

In this paper we study the sequences defined by the last and the last non-zero digits of in base b. For the sequence given by the last digits of in base b, we prove its periodicity using different techniques than those used by W. Sierpinski and R. Hampel. In the case of the sequence given by the last non-zero digits of in base b (which had been studied only for b = 10) we show the non-periodicity of the sequence when b is an odd prime power and when it is even and square-free. We also show that if the sequence is periodic and conjecture that this is the only such case.

In this paper we compute the sum of the k-th powers of all the elements of a finite commutative unital ring, thus generalizing known results for finite fields, the rings of integers modulo n or the ring of Gaussian integers modulo n. As an application, we focus on quotient rings of the form ( Z / n Z ) [ x ] / ( f ( x ) ) for a polynomial f ∈ Z [ x ] .

Generalized Cullen Numbers are positive integers of the form C b ( n ) := n b n + 1 C_b(n):=nb^n+1 . In this work we generalize some known divisibility properties of Cullen Numbers and present two primality tests for this family of integers. The first test is based in the following property of primes from this family: n b n ≡ ( − 1 ) b n^{b^{n}}\equiv (-1)^{b} (mod n b n + 1 nb^n+1 ). It is stronger and has less computational cost than Fermat’s test (to bases b b and n n ) and than Miller-Rabin’s test (if b b is odd, to base n n ). Pseudoprimes for this new test seem to be very scarce, only 4 pseudoprimes have been found among the many millions of Generalized Cullen Numbers tested. We also present a second, more demanding, test for which no pseudoprimes have been found. These tests lead to an algorithm, running in O ~ ( log 2 ⁡ ( N ) ) \tilde {O}(\log ^2(N)) time, which might be very useful in the search of Generalized Cullen Primes.

Here, we show that there is no positive integer n n such that the n n th Cullen number C n = n 2 n + 1 C_n=n2^n+1 has the property that it is composite but ϕ ( C n ) ∣ C n − 1 \phi (C_n)\mid C_n-1 .