7 Rings Definition. A ring consists of a set R on which are defined operations of addition and multiplication satisfying the following axioms: • x+ y = y + x for all elements x and y of R (i.e., addition is commutative); • (x+ y) + z = x+ (y + z) for all elements x, y and z of R (i.e., addition is associative); • there exists an an element 0 of R (known as the zero element) with the property th...

Acknowledgements First of all, I would like to thank my promotor, Paul Van Dooren. It has been a pleasure, beside an honor, to collaborate with him. His knowledge, his enthusiasm and his friendship have always been a great support for me these last four years. He also gave me the opportunity to work with Yves Genin, and for this, I am also very grateful! I can only thank Yves for our collaborat...

In this paper, we investigate the interplay between positive-definite integral ternary quadratic forms and class numbers. We generalize a result of Jones relating theta function for genus form to Hurwitz numbers, obtaining an asymptotic formula (with main term error away from finitely many bad square classes <mml:math xmlns:mml="http://www.w3.org/1998...

We investigate the frequency of positive squareful numbers x, y, z 6 B for which x + y = z and present a conjecture concerning its asymptotic behaviour. Mathematics Subject Classification (2010). 11D45 (11P05, 14G05).

In this paper, we mainly prove the following conjecture of Z.-H. Sun cite{SH20}: Let $p>3$ be a prime. Then $$\sum_{k=0}^{p-1}\binom{2k}k\frac{3k+1}{(-16)^k}f_k\equiv(-1)^{(p-1)/2}p+p^3E_{p-3}\pmod{p^4},$$ where $f_n=\sum_{k=0}^n\binom{n}k^3$ and $E_n$ stand for $n$th Franel number Euler respectively.