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Solved and Unsolved Problems in Number Theory.

Problems on Fibonacci Numbers and Their Generalizations

Following recent papers [1], [5], [22], [23] a composite n is called a pseudoprime to base $b if {{b}^{{n - 1}}} \equiv 1 \bmod n. $

Solved and Unsolved Problems on Pseudoprime Numbers and Their Generalizations

We use the properties of $p$-adic integrals and measures to obtain general congruences for Genocchi numbers and polynomials and tangent coefficients. These congruences are analogues of the usual Kummer congruences for Bernoulli numbers, generalize known congruences for Genocchi numbers, and provide new congruences systems for Genocchi polynomials and tangent coefficients.

On pseudoprimes having special forms and a solution of K. Szymiczek’s problem

LetUn=(αn-β2)/(α-β) forn odd andUn=(αn-βn)/(α2-β2) for evenn, where α and β are distinct roots of the trinomialf(z)=z2-√Lz+Q andL>0 andQ are rational integers.Un is then-th Lehmer number connected withf(z).

Arithmetical progressions formed byk different Lehmer pseudoprimes

A positive integern is called a pseudoprime ifn|2
 n
 −2 andn is composite. W. Sierpinski put forward the following problem: Do there infinitely many arithmetical progressions formed of four pseudoprimes? In this paper it is proved that answer to this problem is in the affirmative.

The solution of W. Sierpiński's problem

An odd compositen is an Euler pseudoprime for the basea if (a, n)=1 and$a^{\left( {n - 1} \right)/2} \equiv \left( {\frac{a}{n}} \right)$ (modn), where$\left( {\frac{a}{n}} \right)$ is the Jacobi symbol. It is shown that for every odda≥3 there exist infinitely many arithmetical progressions consisting of three different Euler pseudoprimes for the basea.

Arithmetical progressions formed from three different Euler pseudoprimes for the odd basea

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The solution of W. Sierpiński's problem

Arithmetical progressions formed from three different Euler pseudoprimes for the odd basea