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

for exponents *n* and *m* greater than one, is finite.^{ [1] }^{ [2] }

The theorem was proven by Dutch number theorist Robert Tijdeman in 1976,^{ [3] } making use of Baker's method in transcendental number theory to give an effective upper bound for *x*,*y*,*m*,*n*. Michel Langevin computed a value of exp exp exp exp 730 for the bound.^{ [1] }^{ [4] }^{ [5] }

Tijdeman's theorem provided a strong impetus towards the eventual proof of Catalan's conjecture by Preda Mihăilescu.^{ [6] } Mihăilescu's theorem states that there is only one member to the set of consecutive power pairs, namely 9=8+1.^{ [7] }

That the powers are consecutive is essential to Tijdeman's proof; if we replace the difference of *1* by any other difference *k* and ask for the number of solutions of

with *n* and *m* greater than one we have an unsolved problem,^{ [8] } called the generalized Tijdeman problem. It is conjectured that this set also will be finite. This would follow from a yet stronger conjecture of Subbayya Sivasankaranarayana Pillai (1931), see Catalan's conjecture, stating that the equation only has a finite number of solutions. The truth of Pillai's conjecture, in turn, would follow from the truth of the abc conjecture.^{ [9] }

In mathematics, a **Diophantine equation** is a polynomial equation, usually involving two or more unknowns, such that the only solutions of interest are the integer ones. 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. The integers 2^{3} and 3^{2} are two powers of natural numbers whose values (8 and 9, respectively) are consecutive. The theorem states that this is the *only* case of two consecutive 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 which, for any given Diophantine equation, can decide whether the equation has a solution with all unknowns taking integer values.

The ** abc conjecture** is a conjecture in number theory, first proposed by Joseph Oesterlé (1988) and David Masser (1985). It is stated in terms of three positive integers,

**Algebraic number theory** is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and function fields. These properties, such as whether a ring admits unique factorization, the behavior of ideals, and the Galois groups of fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations.

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.

In mathematics, **Diophantine geometry** is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and *p*-adic fields. It is a sub-branch of arithmetic geometry and is one approach to the theory of Diophantine equations, formulating questions about such equations in terms of algebraic geometry.

In mathematics, **Roth's theorem** is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half a century, the meaning of *very good* here was refined by a number of mathematicians, starting with Joseph Liouville in 1844 and continuing with work of Axel Thue (1909), Carl Ludwig Siegel (1921), Freeman Dyson (1947), and Klaus Roth (1955).

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. It is named after Srinivasa Ramanujan, who conjectured that it has only five integer solutions, and after Trygve Nagell, who proved the conjecture.

This is a glossary of **arithmetic and diophantine geometry** in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.

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 mathematics, the **subspace theorem** says that points of small height in projective space lie in a finite number of hyperplanes. It is a result obtained by Wolfgang M. Schmidt (1972).

In mathematics, **exponential polynomials** are functions on fields, rings, or abelian groups that take the form of polynomials in a variable and an exponential function.

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, **Szpiro's conjecture** relates the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and Brocard's problem.

In number theory, **Fermat's Last Theorem** states that no three positive integers *a*, *b*, and *c* satisfy the equation *a*^{n} + *b*^{n} = *c*^{n} 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.

In algebraic number theory, **Leopoldt's conjecture**, introduced by H.-W. Leopoldt, states that the p-adic regulator of a number field does not vanish. The p-adic regulator is an analogue of the usual regulator defined using p-adic logarithms instead of the usual logarithms, introduced by H.-W. Leopoldt (1962).

In number theory, the **Fermat–Catalan conjecture** is a generalization of Fermat's last theorem and of Catalan's conjecture, hence the name. The conjecture states that the equation

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*.

- 1 2 Narkiewicz, Wladyslaw (2011),
*Rational Number Theory in the 20th Century: From PNT to FLT*, Springer Monographs in Mathematics, Springer-Verlag, p. 352, ISBN 978-0-857-29531-6 - ↑ Schmidt, Wolfgang M. (1996),
*Diophantine approximations and Diophantine equations*, Lecture Notes in Mathematics,**1467**(2nd ed.), Springer-Verlag, p. 207, ISBN 978-3-540-54058-8, Zbl 0754.11020 CS1 maint: discouraged parameter (link) - ↑ Tijdeman, Robert (1976), "On the equation of Catalan",
*Acta Arithmetica*,**29**(2): 197–209, doi: 10.4064/aa-29-2-197-209 , Zbl 0286.10013 - ↑ Ribenboim, Paulo (1979),
*13 Lectures on Fermat's Last Theorem*, Springer-Verlag, p. 236, ISBN 978-0-387-90432-0, Zbl 0456.10006 CS1 maint: discouraged parameter (link) - ↑ Langevin, Michel (1977), "Quelques applications de nouveaux résultats de Van der Poorten",
*Séminaire Delange-Pisot-Poitou, 17e Année (1975/76), Théorie des Nombres*,**2**(G12), MR 0498426 - ↑ Metsänkylä, Tauno (2004), "Catalan's conjecture: another old Diophantine problem solved" (PDF),
*Bulletin of the American Mathematical Society*,**41**(1): 43–57, doi: 10.1090/S0273-0979-03-00993-5 - ↑ Mihăilescu, Preda (2004), "Primary Cyclotomic Units and a Proof of Catalan's Conjecture",
*Journal für die reine und angewandte Mathematik*,**2004**(572): 167–195, doi:10.1515/crll.2004.048, MR 2076124 - ↑ Shorey, Tarlok N.; Tijdeman, Robert (1986).
*Exponential Diophantine equations*. Cambridge Tracts in Mathematics.**87**. Cambridge University Press. p. 202. ISBN 978-0-521-26826-4. MR 0891406. Zbl 0606.10011. - ↑ Narkiewicz (2011), pp. 253–254

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