8 papers
Two-colorings of finite grids: variations on a theorem of Tibor Gallai
Bogdan Dumitru, Mihai Prunescu
A celebrated but non-effective theorem of Tibor Gallai states that for any finite set of and for any finite number of colors there is a minimal such that no colo…
Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs
Mihai Prunescu
We construct closed forms that generate with repetitions all Mersenne primes, respectively all Fermat primes, all twin-prime pairs and all Sophie Germain primes. Also, we construct…
Elementary closed-forms for non-trivial divisors
Mihai Prunescu, Joseph M. Shunia
We present several elementary closed-forms that express a non-trivial divisor for every composite integer . Each closed-form consists of a fixed number of elementary arithme…
On polynomial systems of equations in square matrices filled with natural numbers
Mihai Prunescu
The positive existential theories of the sets without parameters build an inclusion lattice isomorhic with the lattice of divisibility. All these sets are algorith…
On the first-order theory of the remainder
Mihai Prunescu
It is proved that the first-order theory of the structure (N,mod) is undecidable. Here mod denotes the operation of computing the remainder for any division between positive intege…
On modular representations of C-recursive integer sequences
Mihai Prunescu, Joseph M. Shunia
Prunescu and Sauras-Altuzarra showed that all C-recursive sequences of natural numbers have an arithmetic div-mod representation that can be derived from their generating function.…