* Case g = 1: no points, or C is an elliptic curve and its rational points form a finitely generated abelian group ( Mordell's Theorem, later generalized to the Mordell – Weil theorem ).

proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis.

Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve.

Mordell's theorem on the finite generation of the group of rational points on an elliptic curve is in Chapter 16, and integer points on the Mordell curve in Chapter 26.

His basic work on Mordell's theorem is from 1921 / 2, as is the formulation of the Mordell conjecture.

The proof of Elsholtz and Tao's bound on the number of solutions involves the Bombieri – Vinogradov theorem, the Brun – Titchmarsh theorem, and a system of modular identities, valid when n is congruent to − c or − 1 / c modulo 4ab, where a and b are any two coprime positive integers and c is any odd factor of a + b. For instance, setting a = b = 1 gives one of Mordell's identities, valid when n is 3 ( mod 4 ).

The space of such forms has dimension 1, which means this definition is possible ; and that accounts for the action of Hecke operators on the space being by scalar multiplication ( Mordell's proof of Ramanujan's identities ).

: While others at the time shared this viewpoint ( e. g., Weil, Tate, Serre ), it is easy to forget that others did not, as Mordell's review of Diophantine Geometry attests.

The traditional arrangement of material on Diophantine equations was by degree and number of variables, as in Mordell's Diophantine Equations ( 1969 ).

However, as showed, a polynomial identity that provides a solution for values of n congruent to r mod p can exist only when r is not a quadratic residue modulo p. For instance, 2 is a not a quadratic residue modulo 3, so the existence of an identity for values of n that are congruent to 2 modulo 3 does not contradict Mordell's result, but 1 is a quadratic residue modulo 3 so the result proves that there can be no similar identity for values of n that are congruent to 1 modulo 3.

From Mordell's identities one can conclude that there exists a solution for all n except possibly those that are 1, 121, 169, 289, 361, or 529 modulo 840.

Mordell's book starts with a remark on homogeneous equations f = 0 over the rational field, attributed to C. F. Gauss, that non-zero solutions in integers ( even primitive lattice points ) exist if non-zero rational solutions do, and notes a caveat of L. E. Dickson, which is about parametric solutions.

The Hilbert-Hurwitz result from 1890 reducing the diophantine geometry of curves of genus 0 to degrees 1 and 2 ( conic sections ) occurs in Chapter 17, as does Mordell's conjecture.

Despite Mordell's result limiting the form these congruence identities can take, there is still some hope of using modular identities to prove the Erdős – Straus conjecture.

He had introduced the adele ring in the late 1930s, following Claude Chevalley's lead with the ideles, and given a proof of the Riemann – Roch theorem with them ( a version appeared in his Basic Number Theory in 1967 ).

In 1956, he applied the same thinking to the Riemann – Roch theorem, which had already recently been generalized to any dimension by Hirzebruch.

For a certain class of Green functions coming from solutions of integral equations, Schmidt had shown that a property analogous to the Arzelà – Ascoli theorem held in the sense of mean convergence — or convergence in what would later be dubbed a Hilbert space.

Twenty years earlier, Paul Gordan had demonstrated the theorem of the finiteness of generators for binary forms using a complex computational approach.

In 1976, while other teams of mathematicians were racing to complete proofs, Kenneth Appel and Wolfgang Haken at the University of Illinois announced that they had proven the theorem.

For example, both the Egyptians and the Babylonians were aware of versions of the Pythagorean theorem about 1500 years before Pythagoras ; the Egyptians had a correct formula for the volume of a frustum of a square pyramid ;

Historically, ML stands for metalanguage: it was conceived to develop proof tactics in the LCF theorem prover ( whose language, pplambda, a combination of the first-order predicate calculus and the simply typed polymorphic lambda calculus, had ML as its metalanguage ).

For, in 1823, Abel had at last proved the impossibility of solving the quintic equation in radicals ( now referred to as the Abel – Ruffini theorem ).

Abel had sent most of his work to Berlin to be published in Crelles Journal, but he had saved what he regarded his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials.

Abel gave a proof of the binomial theorem valid for all numbers, extending Euler's result which had held only for rationals.

Germain used this result to prove the first case of Fermat's Last Theorem for all odd primes p < 100, but according to Andrea del Centina, “ she had actually shown that it holds for every exponent p < 197 .” L. E. Dickson later used Germain's theorem to prove Fermat's Last Theorem for odd primes less than 1700.

Such structures had been studied prior to Thurston, but his work, particularly the next theorem, would bring them to prominence.

Since any such axiom system would contain the finitary arithmetic as a subsystem, Gödel's theorem implied that it would be impossible to prove the system's consistency relative to that ( since it would then prove its own consistency, which Gödel had shown was impossible ).

He wrote his first completeness theorem in modal logic at the age of 17, and had it published a year later.

The theorem of quadratic reciprocity ( which he had first succeeded in proving in 1796 ) relates the solvability of the congruence x < sup > 2 </ sup > ≡ q ( mod p ) to that of x < sup > 2 </ sup > ≡ p ( mod q ).

The jury convicted, but the case went to appeal on the basis that no means of accumulating evidence had been provided for jurors who did not wish to use Bayes ' theorem.

The key insight, however, that earned them this credit, was the fundamental theorem of calculus relating differentiation and integration: this rendered obsolete most previous methods for computing areas and volumes, which had not been significantly extended since the time of Ibn al-Haytham ( Alhazen ).

However, the priority for this result ( now known as the Mohr – Mascheroni theorem ) belongs to the Dane Georg Mohr, who had previously published a proof in 1672.

This particular remark of Khayyám and certain propositions found in his Algebra book has made some historians of mathematics believe that Khayyám had indeed a binomial theorem up to any power.

The argument supporting the claim that Khayyám had a general binomial theorem is based on his ability to extract roots.

Earlier statements of this type had either been, except for Gentzen, extremely complicated, ad-hoc constructions ( such as the statements generated by the construction given in Gödel's incompleteness theorem ) or concerned metamathematics or combinatorial results.

It had been realised ( probably by several people ) that failure to complete proofs in the general case of Fermat's last theorem by factorisation using the roots of unity was for a very good reason: a failure of the fundamental theorem of arithmetic to hold in the rings generated by those roots of unity was a major obstacle.

However, the majority are solved via ad hoc methods such as Størmer's theorem or even trial and error.

Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument.

the characteristic polynomial is given by p ( λ )= λ < sup > 2 </ sup >−( a + d ) λ +( ad − bc ), so the Cayley – Hamilton theorem states that

For example, if a bounded differentiable function f defined on a closed interval in the real line has a single critical point, which is a local minimum, then it is also a global minimum ( use the intermediate value theorem and Rolle's theorem to prove this by reductio ad absurdum ).

A semantic proof of Tarski's theorem from Post's theorem is obtained by reductio ad absurdum as follows.

The proof of Tarski's undefinability theorem in this form is again by reductio ad absurdum.

The Lerner symmetry theorem is a result used in trade theory, which states that, based on an assumption of a zero balance of trade ( that is, the value of exported goods equals the value of imported goods for a given country ), an ad valorem import tariff ( a percentage of value or an amount per unit ) will have the same effects as an export tax.

The theorem is based on the observation that the effect on relative prices is the same regardless of which policy ( ad valorem tariffs or export taxes ) is applied.