Diophantine sets are subsets defined by polynomial equations over integers. More precisely We obtain them basically by projecting integers solutions of some polynomial equations. Observation 1: Using the statement iff we see that there is no difference between one polynomial equation and a system of polynomial equations. Example: Even numbers are Diophantine: Odd numbers: Non-negative […]
Category: Number Theory
Mordell’s Equation
The equation is called Mordell’s equation. For , it is non-singular and defines an elliptic curve (genus 1, group structure). Rational Points: We first describe the torsion part of the rational solutions. By Nagell-Lutz, these solutions should have integer coordinates. Torsion: Assume that is free of sixth powers. If not we can use the change […]
Sophie Germain’s Theorem
As early attempts to the proof of Fermat’s last theorem, many mathematicians solved the problem for small exponents. While these special cases are being studied, Sophie Germain, a French mathematician, came up with the following interesting result. (Look at https://www.agnesscott.edu/lriddle/women/germain.htm for her fascinating and revolutionary story) Theorem 1: For any odd prime such that is […]
Kürschâk and Nagel’s theorems (Erdos 1932)
None of the above quantities are integers. Proof: For the first expression, look at the largest prime- when we clear denominators, the denominator is divisible by this prime and numerator is not.For the second expression, if the is smaller then then the quantity is less than one, otherwise there will be a prime between $m$ […]
Betrand Postulate : Erdos( 1932)
The theorem, known as Bertrand’s Postulate, asserts that for any integer , there is always a prime number satisfying . Erdős’s proof (1932) is centered on a careful analysis of the prime factors of the central binomial coefficient, . Contradiction Hypothesis: Assume there exists an integer for which no prime p exists in the interval […]
Zeta(2)
Basel Problem 1644 asks to find the value of Euler (1735) showed that The result, , is a cornerstone of analysis, and there are diverse methods of proof from function theory, harmonic analysis, and geometry. All of them essentially uncover how integers sit inside real numbers. Euler’s Proof: Euler’s original method relies on two representations […]
Eisenstein’s Lattice Point Proof of Quadratic Reciprocity
The Law of Quadratic Reciprocity is a cornerstone of classical number theory—Gauss himself called it the “Theorema Aureum” (Golden Theorem). Although Gauss provided multiple proofs, Eisenstein’s geometric argument simplifies Gauss’s third proof by employing a lattice‐point counting method. Statement of the Law of Quadratic Reciprocity Let and be distinct odd primes. The Law of Quadratic […]
A Neighbourhood Of Infinity 