Does have integer solutions other than ?
Brocard's problem is a problem in mathematics that seeks integer values of such that is a perfect square, where is the factorial. Only three values of are known — 4, 5, 7 — and it is not known whether there are any more.
More formally, it seeks pairs of integers and such that
The problem was posed by Henri Brocard in a pair of articles in 1876 and 1885, [1] [2] and independently in 1913 by Srinivasa Ramanujan. [3]
Pairs of the numbers that solve Brocard's problem were named Brown numbers by Clifford A. Pickover in his 1995 book Keys to Infinity, after learning of the problem from Kevin S. Brown. [4] As of October 2022, there are only three known pairs of Brown numbers:
based on the equalities
Paul Erdős conjectured that no other solutions exist. Computational searches up to one quadrillion have found no further solutions. [5] [6] [7]
It would follow from the abc conjecture that there are only finitely many Brown numbers. [8] More generally, it would also follow from the abc conjecture that
has only finitely many solutions, for any given integer , [9] and that
has only finitely many integer solutions, for any given polynomial of degree at least 2 with integer coefficients. [10]
In mathematics, a Diophantine equation is an equation, typically a polynomial equation in two or more unknowns with integer coefficients, for which only integer solutions are of interest. A linear Diophantine equation equates to a constant the sum of two or more monomials, each of degree one. An exponential Diophantine equation is one in which unknowns can appear in exponents.
Catalan's conjecture (or Mihăilescu's theorem) is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu at Paderborn University. The integers 23 and 32 are two perfect powers (that is, powers of exponent higher than one) of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the only case of two consecutive perfect powers. That is to say, that
Hilbert's tenth problem is the tenth on the list of mathematical problems that the German mathematician David Hilbert posed in 1900. It is the challenge to provide a general algorithm that, for any given Diophantine equation, can decide whether the equation has a solution with all unknowns taking integer values.
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
71 (seventy-one) is the natural number following 70 and preceding 72.
A powerful number is a positive integer m such that for every prime number p dividing m, p2 also divides m. Equivalently, a powerful number is the product of a square and a cube, that is, a number m of the form m = a2b3, where a and b are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers powerful.
The Beal conjecture is the following conjecture in number theory:
In mathematics, in the field of number theory, the Ramanujan–Nagell equation is an equation between a square number and a number that is seven less than a power of two. It is an example of an exponential Diophantine equation, an equation to be solved in integers where one of the variables appears as an exponent.
The Erdős–Straus conjecture is an unproven statement in number theory. The conjecture is that, for every integer that is 2 or more, there exist positive integers , , and for which
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties to the real numbers.
In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations. It follows from the Thue–Siegel–Roth theorem that there are only a finite number of pairs of this type, but Størmer gave a procedure for finding them all.
In number theory, Tijdeman's theorem states that there are at most a finite number of consecutive powers. Stated another way, the set of solutions in integers x, y, n, m of the exponential diophantine equation
In mathematics, the Goormaghtigh conjecture is a conjecture in number theory named for the Belgian mathematician René Goormaghtigh. The conjecture is that the only non-trivial integer solutions of the exponential Diophantine equation
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions.
Trygve Nagell or Trygve Nagel was a Norwegian mathematician, known for his works on Diophantine equations in number theory.
The Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some k-th powers equals the sum of some other k-th powers, then the total number of terms in both sums combined must be at least k.
In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So:
In the mathematics of sums of powers, it is an open problem to characterize the numbers that can be expressed as a sum of three cubes of integers, allowing both positive and negative cubes in the sum. A necessary condition for an integer to equal such a sum is that cannot equal 4 or 5 modulo 9, because the cubes modulo 9 are 0, 1, and −1, and no three of these numbers can sum to 4 or 5 modulo 9. It is unknown whether this necessary condition is sufficient.