Zeros of simultaneous polynomials Sphere and slanted circleIn classical algebraic geometry, the main objects of interest are the vanishing sets of collections of polynomials, meaning the set of all po ...
A perfect cuboid (also called a perfect box) is an Euler brick whose space diagonal also has integer length.In other words, the following equation is added to the system of Diophantine equations defin ...
Let H be a Hadamard matrix of order n. The transpose of H is closely related to its inverse. The correct formula is:H H^{\mathrm{T}} = n I_n \ where In is the n × n identity matrix and HT is the tran ...
The first part of Hilbert's 16th problem In 1876 Harnack investigated algebraic curves in the real projective plane and found that curves of degree n could have no more than{n^2-3n+4 \over 2} separate ...
How to distinguish between different meanings of "algebra"For historical reasons, the word "algebra" has several related meanings in mathematics, as a single word or with qualifiers. Such a situation, ...
Pollock's conjectures are two closely related unproven conjectures in additive number theory. According to Pollock's octahedral numbers conjecture, every positive integer is the sum of at most seven o ...
The Erd?s–Turán conjecture is an old unsolved problem in additive number theory (not to be confused with Erd?s conjecture on arithmetic progressions) posed by Paul Erd?s and Pál Turán in 1941.H ...
Erd?s' conjecture on arithmetic progressions, often referred to as the Erd?s–Turán conjecture due to Turán's earlier work with Erd?s, is a conjecture in arithmetic combinatorics. (not to be conf ...
Gilbreath observed a pattern while playing with the ordered sequence of prime numbers2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, ...Computing the absolute value of the difference between term n+1 and term ...
In mathematics, a diophantine m-tuple is a set of m positive integers \{a_1, a_2, a_3, a_4,\ldots, a_m\} such that a_i a_j + 1 is a perfect square for any 1\le i j \le m.Diophantus himself found the ...
Diophantine equations, such as the integer version of the equation a2 + b2 = c2 that appears in the Pythagorean theorem, have been studied for their integer solution properties for centuries. Fermat's ...
Consider the following operation on an arbitrary positive integer:If the number is even, divide it by two.If the number is odd, triple it and add one.In modular arithmetic notation, define the functio ...
Relationship with Lagrange's four-square theorem Long before Waring posed his problem, Diophantus had asked whether every positive integer could be represented as the sum of four perfect squares great ...
In number theory, Goldbach's weak conjecture, also known as the odd Goldbach conjecture, the ternary Goldbach problem, or the 3-primes problem, states that:Every odd number greater than 5 can be expre ...
A Goldbach number is an even positive integer that can be expressed as the sum of two primes. Therefore, another statement of Goldbach's conjecture is that all even integers greater than or equal to 4 ...
