# a lemma upon the Surface Loci of Euclid which states that the locus of a point such that its distance from a given point bears a constant ratio to its distance from a given straight line is a conic, and is followed by proofs that the conic is a parabola, ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than 1 ( the first recorded proofs of the properties, which do not appear in Apollonius ).

The difficulty is in effective computation of bounds: for a given discriminant, it is easy to compute the class number, and there are several ineffective lower bounds on class number ( meaning that they involve a constant that is not computed ), but effective bounds ( and explicit proofs of completeness of lists ) are harder.

This fact can be used to give proofs of the Brouwer fixed point theorem and the Borsuk – Ulam theorem in dimension 2.

The various invalid mathematical proofs ( e. g., that 1 = 2 ) are classic examples, generally relying on a hidden division by zero.

( See square root of 2 for proofs that this is an irrational number, and quadratic irrational for a proof for all non-square natural numbers.

In 1992 S. V. Ivanov announced the negative solution for sufficiently large even exponents divisible by a large power of 2 ( detailed proofs were published in 1994 and occupied some 300 pages ).

Level 2: Theorem provers may be used to undertake fully formal machine-checked proofs.

It is possible to disguise a special case of division by zero in an algebraic argument, leading to spurious proofs that 1 = 2 such as the following:

There are also many deep and wide ranging conjectures in number theory whose proofs seem too difficult for current techniques, such as the Twin prime conjecture which asks whether there are infinitely many primes p such that p + 2 is prime.

Unless specified otherwise, correct proofs should always be accepted, and incorrect proofs should be rejected with probability greater than 1 / 2.

Schubert's last task in life was the correction of the proofs for part 2 of Winterreise, and his thoughts while correcting those of the last song, " Der Leiermann ", when his last illness was only too evident, can only be imagined.

gives an example of a combinatorial enumeration problem ( counting the number of sequences of k subsets S < sub > 1 </ sub >, S < sub > 2 </ sub >, ... S < sub > k </ sub >, that can be formed from a set of n items such that the subsets have an empty common intersection ) with two different proofs for its solution.

After comparing what writings he could find, Erasmus wrote corrections between the lines of the manuscripts he was using ( among which was Minuscule 2 ) and sent them as proofs to Froben.

The " Fons Vitæ " consists of five tractates, treating respectively of ( 1 ) matter and form in general and their relation in physical substances (" substantiæ corporeæ sive compositæ "); ( 2 ) the substance which underlies the corporeality of the world (" de substantia quæ sustinet corporeitatem mundi "); ( 3 ) proofs of the existence of " substantiæ simplices ", of intermediaries between God and the physical world ; ( 4 ) proofs that these " substantiæ simplices ", or " intelligibiles ", are likewise constituted of matter and form ; ( 5 ) universal matter and universal form.

" During the course of the talk he outlined proofs that ζ ( 3 ) and ζ ( 2 ) were irrational, the latter using methods simplified from those used to tackle the former rather than relying on the expression in terms of π.

Elvo Zornitta required 2, 500, 000 euro as reparation for the manipulated proofs.

2 ) Descartes ' proofs of God's existence presuppose the reliability of clear and distinct perceptions.

2 of 1933 while in printer's galley proofs.

*< cite id = refMott1729b > Andrew Motte ( 1729b ) ( translator ), " The Mathematical Principles of Natural Philosophy, by Sir Isaac Newton, translated into English ", Volume II, containing Books 2 and 3 ( with Index, Appendix containing additional ( Newtonian ) proofs, and " The Laws of the Moon's Motion according to Gravity ", by John Machin ).

2: 185 ) The month of Ramadhan is that in which the Qur ' an was revealed, a guidance to men and clear proofs of the guidance and the distinction ; therefore, whoever of you is present in the month, he shall fast therein, and whoever is sick or upon a journey, then ( he shall fast ) a ( like ) number of other days ; God desires ease for you, and does not desire for you difficulty, and ( desires ) that you should complete the number and that you should exalt the greatness of God for having guided you and that you may give thanks.

As many other important theorems, the Second Main Theorem has several different proofs.

In 1942, he provided one of the first proofs of the First and Second Welfare Theorems.

A more successful effort was the Standard Proof for Pythagoras ' Theorem, that replaced the more than 100 incompatible existing proofs.

There are now several different proofs of Perelman's Theorem 7. 4, or variants of it which are sufficient to prove geometrization.

Also containing proofs of Perelman's Theorem 7. 4, there is a paper of Morgan and Tian, another paper of Kleiner and Lott, and a paper by Cao and Ge.

His contributions were plentiful, including the characterization of binary, regular, and graphic matroids by excluded minors ; the regular-matroid representability theorem ; the theory of chain groups and their matroids ; and the tools he used to prove many of his results, the " Path Theorem " and " Homotopy Theorem " ( see, e. g., ), which are so complex that later theorists have gone to great trouble to eliminate the necessity of using them in proofs.

The non-solvability of in integers is sufficient to show the non-solvability of in integers, which is a special case of Fermat's Last Theorem, and the historical proofs of the latter proceeded by more broadly proving the former using infinite descent.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

As discussed above, in ZFC, the axiom of choice is able to provide " nonconstructive proofs " in which the existence of an object is proved although no explicit example is constructed.

In propositional logic, associativity is a valid rule of replacement for expressions in logical proofs.

For theoretical analysis, this approach is more suited for constructing detailed formal proofs and is generally preferred in the research literature.

Another way of making the point is that if the Platonic world were to disappear, it would make no difference to the ability of mathematicians to generate proofs, etc., which is already fully accountable in terms of physical processes in their brains.

* Metamath-a language for developing strictly formalized mathematical definitions and proofs accompanied by a proof checker for this language and a growing database of thousands of proved theorems ; while the Metamath language is not accompanied with an automated theorem prover, it can be regarded as important because the formal language behind it allows development of such a software ; as of March, 2012, there is no " widely " known such software, so it is not a subject of " automated theorem proving " ( it can become such a subject ), but it is a proof assistant.

Using programs or proofs of bounded lengths, it is possible to construct an analogue of the Berry expression in a formal mathematical language, as has been done by Gregory Chaitin.

Sometimes a conjecture is called a hypothesis when it is used frequently and repeatedly as an assumption in proofs of other results.

These " proofs ", however, would fall apart if it turned out that the hypothesis was false, so there is considerable interest in verifying the truth or falsity of conjectures of this type.

In simple terms, co-NP is the class of problems for which efficiently verifiable proofs of no instances, sometimes called counterexamples, exist.

A famous network of conditional proofs is the NP-complete class of complexity theory.

The result can be considered as a type of generalized geometry, projective geometry, but it can also be used to produce proofs in ordinary Euclidean geometry in which the number of special cases is reduced.

# The fatwā is in line with relevant legal proofs, deduced from Qur ' anic verses and ahadith ; provided the hadith was not later abrogated by Muhammad.

There have been many other arguments against ontological proofs such as: Existence precedes essence ; Gaunilo's island ; Necessary nonexistence ; Existence is not a predicate ; and Problem of incoherence.

Such proofs presume the existence of a totality that is complete, a notion disallowed by intuitionists when extended to the infinite — for them the infinite can never be completed:

While it is one of the most commonly used concepts in logic it must not be mistaken for a logical law ; rather, it is one of the accepted mechanisms for the construction of deductive proofs that includes the " rule of definition " and the " rule of substitution " Modus ponens allows one to eliminate a conditional statement from a logical proof or argument ( the antecedents ) and thereby not carry these antecedents forward in an ever-lengthening string of symbols ; for this reason modus ponens is sometimes called the rule of detachment.