He uses this as a proof of the existence of atoms:

In this case, if a proof uses this statement, researchers will often look for a new proof that doesn't require the hypothesis ( in the same way that it is desirable that statements in Euclidean geometry be proved using only the axioms of neutral geometry, i. e. no parallel postulate.

Empire magazine praised the film saying " the gaudily gory, virtuoso, hyper-kinetic horror sequel / remake uses every trick in the cinematic book " and confirms that " Bruce Campbell and Raimi are gods " and Caryn James of The New York Times called it " genuine, if bizarre, proof of Sam Raimi's talent and developing skill.

The proof uses modal logic, which distinguishes between necessary truths and contingent truths.

The proof of Gödel's completeness theorem given by Kurt Gödel in his doctoral dissertation of 1929 ( and a rewritten version of the dissertation, published as an article in 1930 ) is not easy to read today ; it uses concepts and formalism that are outdated and terminology that is often obscure.

George Boolos has since sketched an alternative proof of the first incompleteness theorem that uses Berry's paradox rather than the liar paradox to construct a true but unprovable formula.

The proof of this statement uses the linearity of classical probability, and has exactly the same structure as the proof of the quantum no-cloning theorem.

Newman's proof is arguably the simplest known proof of the theorem, although it is non-elementary in the sense that it uses Cauchy's integral theorem from complex analysis.

Perelman's proof uses a modified version of a Ricci flow program developed by Richard Hamilton.

The proof uses the classification of finite simple groups.

Although BB84 does not use entanglement, Ekert's protocol uses the violation of a Bell's inequality as a proof of security.

The proof of Euler's identity uses only the formula for the geometric series and the fundamental theorem of arithmetic.

A sig block is easily forged, whereas a digital signature uses cryptographic techniques to provide verifiable proof of authorship.

* Centipede game, the Nash equilibrium of which uses a similar mechanism as its proof.

The alternative is formal software verification, which uses mathematical proof techniques to show the absence of bugs.

An alternative proof uses Prüfer sequences.

While the proof that is a Noetherian ring uses the order structure of, typical proofs in ring theory in general do not assume such additional structure on the ring.

One proof of the impossibility of finding a planar embedding of K < sub > 3, 3 </ sub > uses a case analysis involving the Jordan curve theorem, in which one examines different possibilities for the locations of the vertices with respect to the 4-cycles of the graph and shows that they are all inconsistent with a planar embedding.

The original proof uses hypergeometric functions and innovative tools from the theory of Hilbert spaces of entire functions, largely developed by de Branges.

In logic, a formal proof is not written in a natural language, but instead uses a formal language consisting of certain strings of symbols from a fixed alphabet.

This proof uses definition of even integers, as well as distribution law.

Euclid's original proof adds a third: the two lengths are not prime to one another.

A proof from Euclid's elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides.

The modern formulation of proof by induction was not developed until the 17th century, but some later commentators consider it implicit in some of Euclid's proofs, e. g., the proof of the infinitude of primes.

Book VII, propositions 30 and 32 of Euclid's Elements is essentially the statement and proof of the fundamental theorem.

A proof from Euclid | Euclid's Euclid's Elements | Elements, widely considered the most influential textbook of all time.

The earliest implicit traces of mathematical induction can be found in Euclid's proof that the number of primes is infinite and in Bhaskara's " cyclic method ".

#* Note: This fact provides a proof of the infinitude of primes distinct from Euclid's Theorem: if there were finitely many primes, with p being the largest, we reach an immediate contradiction since all primes dividing 2 < sup > p </ sup > − 1 must be larger than p .</ li >

Euclid's construction for proof of the triangle inequality for plane geometry.

This proof appears in Euclid's Elements, Book 1, Proposition 20.

Euclid's commentator Proclus gave a proof of this postulate using the previous postulates, but it may be argued that this proof makes use of some hidden assumptions.

Because if not, then an elementary proof of Euclid's result is also impossible.

A proof from Euclid's Elements that, given a line segment, an equilateral triangle exists that includes the segment as one of its sides.

However, Euclid's original proof of this proposition is general, valid, and does not depend on the figure used as an example to illustrate one given configuration.

The original proof of the irrationality of the non-square natural numbers depends on Euclid's lemma.

His proof is in Euclid's Elements Book X Proposition 9.

In addition to his significant contributions to number theory algorithms for multiprecision integers, such as factoring, Euclid's algorithm, long division, and proof of primality, he also formulated Lehmer's conjecture and participated in the Cunningham project.

It is widely believed, but false, that the idea of primorial primes appears in Euclid's proof of the infinitude of the prime numbers: First, assume that the first n primes are the only primes that exist.

In fact, Euclid's proof did not assume that a finite set contains all primes that exist.

The pons asinorum in Byrne's edition of the Elements showing part of Euclid's proof.

The easiest proof of Euclid's lemma uses another lemma called Bézout's identity.

The logic of this proof is basically Euclid's, but the notation and some of the concepts ( zero, negative ) would be foreign to him.