arXiv subjects

Samer Seraj

Publications and source records attributed to Samer Seraj.

6 recordsLinked to original sources

arXiv2025

Diophantine Analysis of a Digital Anomaly

The arithmetic-digital anomaly of $5\div 2 = 2.5$ has been observed several times in the past. We generalize it to an exponential Diophantine equation and inequality in the general number base, which is the object of our analysis. First, we produce a near-parametrization of all solutions using a modification of the standard parametrization of Pythagorean triples. We use this parametrized function to find all solutions where the numerator and denominator are coprime, and we construct infinite families where they are not coprime. Next, we use a variant of Baker's theorem from transcendental number theory to prove that each number base admits only finitely many solutions. Lastly, we use the $abc$ conjecture to conditionally show that only finitely many solutions have a numerator with $k$ digits, for each $k\ge 3$. A conjecture is offered for $k=2$.

math.HO↗

arXiv2025

Polymorphic Numbers with Exponential Prefix

We observe that the computation $5^2 = 25$ has the digital property of the result being equal to the exponent concatenated directly to the left of the base. The generalization to a Diophantine equation and inequality in number bases has been articulated previously, but a comprehensive answer was not available in the literature. We classify and largely parametrize the solutions. Tools that play key roles are the Newton-Raphson method, the arithmetic-geometric means inequality, Pell's equation, and Fermat's little theorem.

math.HO↗

arXiv2023

Sum of Distinct Biquadratic Residues Modulo Primes

Two conjectures, posed by Finch-Smith, Harrington, and Wong in a paper published in Integers in $2023$, are proven. Given a monic biquadratic polynomial $f(x) = x^4 + cx^2 + e$, we prove a formula for the sum of its distinct outputs modulo any prime $p\ge 7$. Here, $c$ is an integer not divisible by $p$ and $e$ is any integer. The formula splits into eight cases, depending on the remainder of $p$ modulo $8$ and whether $c$ is a quadratic residue modulo $p$. The formula quickly extends to the non-monic case. We then apply the formula to prove a classification of the set of such sums in terms of the sets of squares and fourth powers, when $c$ in $x^4 + cx^2$ is varied over all integers with a fixed prime modulus $p\ge 7$. The sum and the set of sums are manually computed for the excluded prime moduli $p=3,5$.

math.NT↗

arXiv2021

An Idempotent Cryptarithm

Notice that the square of $9376$ is $87909376$ which has as its rightmost four digits $9376$. To generalize this remarkable fact, we show that, for each integer $n\ge 2$, there exists at least one and at most two positive integers $x$ with exactly $n$-digits in base-$10$ (meaning the leftmost or $n^{\text{th}}$ digit from the right is non-zero) such that squaring the integer results in an integer whose rightmost $n$ digits form the integer $x$. We then generalize the argument to prove that, in an arbitrary number base $B\ge 2$ with exactly $m$ distinct prime factors, an upper bound is $2^m -2$ and a lower bound is $2^{m-1}-1$ for the number of such $n$-digit positive integers. For $n=1$, there are exactly $2^m -1$ solutions, including $1$ and excluding $0$.

math.HO↗

arXiv2021

Functions with Diffusive Properties

While exploring desirable properties of hash functions in cryptography, the author was led to investigate three notions of functions with scattering or "diffusive" properties, where the functions map between binary strings of fixed finite length. These notions of diffusion ask for some property to be fulfilled by the Hamming distances between outputs corresponding to pairs of inputs that lie on the endpoints of edges of an $n$-dimensional hypercube. Given the dimension of the input space, we explicitly construct such functions for every dimension of the output space that allows for the functions to exist.

cs.IT↗

arXiv2013

Sum of Cubes is Square of Sum

Inspired by the fact that the sum of the cubes of the first $n$ naturals is equal to the square of their sum, we explore, for each $n$, the Diophantine equation representing all non-trivial sets of $n$ integers with this property. We find definite answers to the standard question of infinitude of the solutions as well as several other surprising results.

math.NT↗