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Mihai Prunescu

Publications and source records attributed to Mihai Prunescu.

At least 19 recordsLinked to original sources
arXiv2026

Short arithmetic terms express the cardinality of elliptic curves in Weierstra{\ss} normal form over finite fields

Arithmetic terms are fixed finite compositions of additions, multiplications, subtractions, divisions with remainder and integer exponentiations. An arithmetic term in natural numbers $A$, $B$, $n$, obtained by refining the general method of the second author (arXiv:2608.22049) for elliptic curves in Weierstra{\ss} normal form, counts the solutions in $(\mathbb Z/n\mathbb Z)^2$ for every modulus $n \geq 1$, with intermediate integers of approximately $2n^5$ binary digits instead of approximately $2n^{11}$. For a prime modulus $p \geq 17$, a second construction, based on the Hasse invariant and on the trace of Frobenius, gives a term of about thirty operations, in place of the fifty-one products of generalized geometric progressions of the general method.

math.NT↗
arXiv2026

How many points does an affine algebraic set have in residue classes modulo n ?

We show that for every uniform family of affine algebraic sets, there is a formula in the parameters of the family and in $n$, expressing the cardinality of the set in the ring $\mathbb Z/n \mathbb Z$. In particular, we construct such a formula for the elliptic curves in Weierstrass normal form. These formulas are arithmetic terms: fixed finite compositions of additions, multiplications, subtractions, divisions with remainder and integer exponentiations.

math.NT↗
arXiv2026

Undecidability, Chaos and Universality in Arithmetic Terms

Arithmetic terms are finite fixed compositions of additions, subtractions, multiplications, divisions with remainder and exponentiations, containing variables interpreted as natural numbers. They build a well-defined notion of closed formula. It is known that every Kalmar elementary function can be expressed as an arithmetic term. In this paper, one studies the power of expression of the arithmetic terms. By interpreting Hilbert's Tenth Problem in arithmetic terms, it is shown that it is undecidable whether one-variable arithmetic term takes the value $0$, or whether two such terms take or not the same values. An algorithm constructs the arithmetic term representing an arbitrary function, which has been defined by a recurrence rule. This construction has various applications. Functions with chaotic behavior, like the Logistic Map, can be expressed as arithmetic terms. Finally, we construct a Turing complete arithmetic term and we express a Turing universal function by an arithmetic term. A somewhat unexpected application: there is a {\it wise} arithmetic term. It gets (the code of) a sentence, (the code of) a formalized theory and a bound $B$, and after performing a constant number of operations, it outputs (the code of) a proof of the sentence using the given theory if such a proof does exist and its length is less than $B$. Otherwise, it outputs $0$.

math.LO↗
arXiv2025

Two-colorings of finite grids: variations on a theorem of Tibor Gallai

A celebrated but non-effective theorem of Tibor Gallai states that for any finite set $A$ of $\Z^n$ and for any finite number of colors $c$ there is a minimal $m$ such that no coloring of the finite $m^n$-grid can avoid that a homothetic image of $A$ is monochromatic. We find (or confirm) $m$ for equilateral triangles, squares, and various types of rectangles. Also, we extend the problem from homothety to general similarity, or to similarity generated using some special rotations. In particular, we compute Gallai similarity numbers for lattice rectangles similar to $1\times k$ (in all orientations) for $k=2,3,4$. The solutions have been found in the framework of the Satisfiability Problem in Propositional Logic (SAT). While some questions were solved using managed brute force, for the more computationally intensive questions we used modern SAT solvers together with symmetry breaking techniques. Some other minor questions are solved for triangles and squares, and new lower bounds are found for regular hexagons on the triangular lattice and for three-dimensional cubes in $\Z^3$.

math.CO↗
arXiv2025

Arithmetic closed forms count the Mersenne primes, the Fermat primes and the twin-prime pairs

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 closed forms that count all Mersenne primes between $0$ and $2^{n+2}-1$, respectively all Fermat primes between $0$ and $6n+5$ and all twin-prime pairs between $0$ and $n$. Every closed form is an arithmetic term, i. e. a fixed finite composition of the following arithmetic operations: addition, subtraction, multiplication, division with remainder and the exponentiation $2^n$. While for generating these sets with repetitions, only Wilson's Theorem is applied, for the counting forms we use more specific tests, i.e. Lucas-Lehmer, respectively Pepin, and we apply to some extent Jones' work (see Acta Arithmetica XXXV, pg. 210 - 221, 1979). To count twin primes we apply Clement's Theorem, which is closely related to Wilson's. A closed form to count the Sophie Germain primes can be constructed similarly.

math.NT↗
arXiv2025

Elementary closed-forms for non-trivial divisors

We present several elementary closed-forms that express a non-trivial divisor for every composite integer $n > 1$. Each closed-form consists of a fixed number of elementary arithmetic operations drawn from the set: addition, subtraction, multiplication, integer division, and exponentiation. Two families of closed-forms are developed. First, direct application of the hypercube method yields closed-forms $T_1(n)$, $T_2(n)$, $T_3(n)$, and $T_4(n)$ expressing the smallest prime divisor, largest non-trivial divisor, largest prime divisor, and greatest prime $\leq n$, respectively. The factorial-unwinding technique underlying these hypercube constructions leads to extreme symbolic complexity, motivating our main result: An alternative closed-form $T(n)$ that avoids factorial-unwinding by synthesizing the quadratic residue invariants $\chi(n)$ (largest $r$ such that $r^2$ is a divisor) and $\omega(n)$ (number of distinct prime divisors) with integer root extraction. Although evaluating these closed-forms requires exponential time, the number of arithmetic operations performed remains constant and independent of the input size $n$. This sharply contrasts with traditional algorithmic methods, where the number of operations required to locate a non-trivial divisor necessarily scales with $n$.

math.NT↗
arXiv2025

On polynomial systems of equations in square matrices filled with natural numbers

The positive existential theories of the sets $M_n(\mathbb N)$ without parameters build an inclusion lattice isomorhic with the lattice of divisibility. All these sets are algorithmically undecidable. In further sections some easier observations are made, like the undecidability of Diophantine equations with coefficients in $M_n(\mathbb Z)$.

math.LO↗
arXiv2025

Proof verification by polynomial Fingerprinting

To cater to the needs of fast verification for mathematical proofs, we describe a method to encode formal sentences in $2 \times 2$ - matrices over multivariate polynomials with integer coefficients. This correspondence is homomorphic: usual proof-steps like modus-ponens or variable substitution in terms and formulae become operations with matrices. By evaluating the polynomial variables in random elements of a suitably chosen finite field, the proof is replaced by a numeric sequence. Only the values corresponding to axioms and tautologies have to be computed from scratch. The values corresponding to derived formulas are computed from the values corresponding to their ancestors by applying the homomorphic properties. The polynomial matrix corresponding to the conclusion of the proof is also evaluated in the chosen random values. If the last term of the numeric sequence equals the evaluation of the conclusion, by the Schwartz-Zippel Lemma, the proof is with high probability correct.

math.LO↗
arXiv2025

On the first-order theory of the remainder

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 integers; i.e. x mod y is the remainder obtained by the division x : y.

math.LO↗
arXiv2025

A Minimal Substitution Basis for the Kalm\'ar Elementary Functions

We show that the class of Kalm\'ar elementary functions can be inductively generated from the addition, the integer remainder, and the base-two exponentiation, hence improving previous results by Marchenkov and Mazzanti. We also prove that the substitution basis defined by these three operations is minimal. Furthermore, we discuss alternative substitution bases under arity constraints.

math.LO↗
arXiv2025

On modular representations of C-recursive integer sequences

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. This representation consists of computing the quotient of two exponential polynomials and taking the remainder of the result modulo a third exponential polynomial, and works for all integers $n \geq 1$. Using a different approach, Prunescu proved the existence of two other representations, one of which is the mod-mod representation, consisting of two successive remainder computations. This result has two weaknesses: (i) the representation works only ultimately, and (ii) a correction term must be added to the first exponential polynomial. We show that a mod-mod representation without inner correction term holds for all integers $n \geq 1$. This follows directly from the div-mod representation by an arithmetic short-cut from outside.

math.NT↗
arXiv2024

On non-holonomicity, transcendence and $p$-adic valuations

Let ${\nu}_q(n)$ be the p-adic valuation of $n$. We show that the power series with coefficients ${\nu}_q(n)$, respectively ${\nu}_p(n)(\mathrm{ mod\;} k)$, are non-holonomic and not algebraic in characteristic 0. We find infinitely many rational numbers and infinitely many algebraic irrational numbers for which the values of these series are transcendental. We apply these results to some $p$-automatic sequences, one of them being the period-doubling sequence.

math.NT↗
arXiv2024

On arithmetic terms expressing the prime-counting function and the n-th prime

We present the first fixed-length elementary closed-form expressions for the prime-counting function, $\pi(n)$, and the $n$-th prime number, $p(n)$. These expressions are arithmetic terms, requiring only a finite and fixed number of elementary arithmetic operations from the set: addition, subtraction, multiplication, integer division, and exponentiation. Mazzanti proved that every Kalmar function can be represented as an arithmetic term. We develop an arithmetic term representing the prime omega function, $\omega(n)$, which counts the number of distinct prime divisors of a positive integer $n$. From this term, we find immediately an arithmetic term for the prime-counting function, $\pi(n)$. Combining these results with a new arithmetic term for binomial coefficients and novel prime-related exponential Diophantine equations, we manage to develop an arithmetic term for the $n$-th prime number, $p(n)$, thereby providing a constructive solution to the fundamental question: Is there an order to the primes?

math.NT↗
arXiv2024

On the representation of number-theoretic functions by arithmetic terms

We present closed forms for several functions that are fundamental in number theory and we explain the method used to obtain them. Concretely, we find formulas for the p-adic valuation, the number-of-divisors function, the sum-of-divisors function, Euler's totient function, the modular inverse, the integer part of the root, the integer part of the logarithm, the multiplicative order and the discrete logarithm. Although these are very complicated, they only involve elementary operations, and to our knowledge no other closed form of this kind is known for the aforementioned functions.

math.NT↗
arXiv2024

On other two representations of the C-recursive integer sequences by terms in modular arithmetic

An integer sequence that is defined by initial values and a linear recurrence with constant integer coefficients, can be represented by the difference of two arithmetic terms containing exponentiation. All constants occuring in the term are integers. While in the paper "On the representation of C-recursive integer sequences by arithmetic terms" by Prunescu and Sauras-Altuzarra, the terms consist of the remainder operation, applied on a division; the representations shown here are a division applied to a remainder operation, respectively the composition of two remainder operations.

math.NT↗
arXiv2024

On the representation of C-recursive integer sequences by arithmetic terms

We show that, if an integer sequence is given by a linear recurrence of constant rational coefficients, then it can be represented as the difference of two arithmetic terms with exponentiation, which do not contain any irrational constant. We apply our methods to various Lucas sequences including the classical Fibonacci sequence, to the sequence of solutions of the Pell equation and to some natural C-recursive sequences of degree 3.

math.LO↗
arXiv2023

Symmetric Functions over Finite Fields

The number of linear independent algebraic relations among elementary symmetric polynomial functions over finite fields is computed. An algorithm able to find all such relations is described. It is proved that the basis of the ideal of algebraic relations found by the algorithm consists of polynomials having coefficients in the prime field F_p.

cs.SC↗