* Java applet: Factorization using the Elliptic Curve Method to find very large composites

* Factorization using the Elliptic Curve Method, a Java applet which uses ECM and switches to the Self-Initializing Quadratic Sieve when it is faster.

* Distributed computing project yoyo @ Home Subproject ECM is a program for Elliptic Curve Factorization which is used by a couple of projects to find factors for different kind of numbers.

Factorization is thought to be a computationally difficult problem, whereas primality testing is comparatively easy ( its running time is polynomial in the size of the input ).

* Factorization Source Code: by Paul Herman & Ami Fischman, C ++ source code for many factorization algorithms including Pollard Rho & Shor's.

* Fermat's Last Theorem Blog: Unique Factorization, A blog that covers the history of Fermat's Last Theorem from Diophantus of Alexandria to the proof by Andrew Wiles.

Factorization over a finite field behaves similarly to factorization over the rational or the complex field.

Lenat claimed that the system had rediscovered both Goldbach's conjecture and the Unique Prime Factorization Theorem.

* Fast Prime Factorization Calculator in JavaScript.

* Thorsten Kleinjung, Kazumaro Aoki, Jens Franke, Arjen Lenstra, Emmanuel Thomé, Joppe Bos, Pierrick Gaudry, Alexander Kruppa, Peter Montgomery, Dag Arne Osvik, Herman te Riele, Andrey Timofeev and Paul Zimmermann, " Factorization of a 768-bit RSA modulus ", pp. 333-350 in: T. Rabin ( ed.

* IFSSA ( Integer Factorization Signature Scheme with Appendix ): Includes two variants of RSA, Rabin-Williams, and ESIGN, with several message encoding methods.

* P Plus 1 Factorization Method, MersenneWiki.

In particular, the DFT is widely employed in signal processing and related fields to analyze the frequencies contained in a sampled signal, to solve partial differential equations, and to perform other operations such as convolutions or multiplying large integers.

This huge improvement made many DFT-based algorithms practical ; FFTs are of great importance to a wide variety of applications, from digital signal processing and solving partial differential equations to algorithms for quick multiplication of large integers.

1850 BCE " contains fifteen Pythagorean triples with quite large entries, including ( 13500, 12709, 18541 ) which is a primitive triple ,< ref > Three positive integers form a primitive Pythagorean triple if and if the highest common factor of is 1.

For any positive integers and, there exists a graph with girth at least and chromatic number at least ; for instance, the Grötzsch graph is triangle-free and has chromatic number 4, and repeating the Mycielskian construction used to form the Grötzsch graph produces triangle-free graphs of arbitrarily large chromatic number.

Many cryptographic protocols are based on the difficulty of factoring large composite integers or a related problem, the RSA problem.

The most difficult integers to factor in practice using existing algorithms are those that are products of two large primes of similar size, and for this reason these are the integers used in cryptographic applications.

** Knuth's up-arrow notation, a method of notation of very large integers

Informally speaking, the prime number theorem states that if a random integer is selected in the range of zero to some large integer N, the probability that the selected integer is prime is about 1 / ln ( N ), where ln ( N ) is the natural logarithm of N. For example, among the positive integers up to and including N = 10 < sup > 3 </ sup > about one in seven numbers is prime, whereas up to and including N = 10 < sup > 10 </ sup > about one in 23 numbers is prime ( where ln ( 10 < sup > 3 </ sup >)= 6. 90775528. and ln ( 10 < sup > 10 </ sup >)= 23. 0258509 ).

Its security is connected to the ( presumed ) extreme difficulty of factoring large integers, a problem for which there is no known efficient ( i. e. practicably fast ) general technique.

The word size, 12 bits, is large enough to handle unsigned integers from 0 to 4095 – wide enough for controlling simple machinery.

Integer factorization is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers ( e. g., products of two 300-digit primes ).

Quantization is the procedure of constraining something from a relatively large or continuous set of values ( such as the real numbers ) to a relatively small discrete set ( such as the integers ).

RSA is an algorithm for public-key cryptography that is based on the presumed difficulty of factoring large integers, the factoring problem.

For large n, 3 / 4 of the positive integers less than n are not divisible by 4, 8 / 9 of these numbers are not divisible by 9, and so on.

He used it to prove that there exist infinitely many integers n such that n and n + 2 have at most nine prime factors ( 9-almost primes ); and that all large even integers are the sum of two 9 ( or smaller )- almost primes.

Statistical considerations which focus on the probabilistic distribution of prime numbers present informal evidence in favour of the conjecture ( in both the weak and strong forms ) for sufficiently large integers: the greater the integer, the more ways there are available for that number to be represented as the sum of two or three other numbers, and the more " likely " it becomes that at least one of these representations consists entirely of primes.

* Montgomery reduction method for multiplying large integers in modulo field

The fundamental property of used in the proof above is that there cannot be a chain of positive integers where each integer in the chain is strictly less than its predecessor ; in other words, the ring of integers is not " too large " since it cannot sustain such a " large chain ".

To create RSA signature keys, generate an RSA key pair containing a modulus N that is the product of two large primes, along with integers e and d such that e d ≡ 1 ( mod φ ( N )), where φ is the Euler phi-function.

Had the situation been reversed, had, for instance, England been the enemy in 1898 because of issues of concern chiefly to New England, there is little doubt that large numbers of Southerners would have happily put on their old Confederate uniforms to fight as allies of Britain.

It became the sole `` subject '' of `` international law '' ( a term which, it is pertinent to remember, was coined by Bentham ), a body of legal principle which by and large was made up of what Western nations could do in the world arena.

If a work is divided into several large segments, a last-minute drawing of random numbers may determine the order of the segments for any particular performance.

On the one hand, he does not work for a large agency, but is almost always self-employed.

Although we continue to pay our conversational devotions to `` free private enterprise '', `` individual initiative '', `` the democratic way '', `` government of the people '', `` competition of the marketplace '', etc., we live rather comfortably in a society in which economic competition is diminishing in large areas, bureaucracy is corroding representative government, technology is weakening the citizen's confidence in his own power to make decisions, and the threat of war is driving him economically and physically into the ground ''.

Third, the United States is pressing forward in the development of large rocket engines to place vehicles of many tons into space for exploration purposes.

In this domain the simple fact of coexistence in the same local, national, and world community is enough to guarantee that we cannot refrain from having some effect, large or small, upon Gentile-Jewish relations.

This is in large part a code of behavior and a glossary of values: what is it that people do and should do and how one should regard it.

If we are to believe the list of titles printed in Malraux's latest book, La Metamorphose Des Dieux, Vol. 1 ( ( 1957 ), he is still engaged in writing a large novel under his original title.

But it would greatly strengthen any Mayor's executive powers, remove the excuse in large degree that he is a captive of inaction in the Board of Estimate, increase his budget-making authority both as to expense and capital budgets, and vest in him the right to reorganize city departments in the interest of efficiency and economy.

We believe that autism, like so many other conditions of defect and deviation, is to a large extent inborn.

Decca is not the only large commercial company to impart instruction.

For the first time in history the entire world is dominated by two large, powerful nations armed with murderous nuclear weapons that make conventional warfare of the past a nullity.

In any event, the yearly sacrifice of 40,000 victims is a hecatomb too large to be justified by the most ardent faith.

The lack of scientific unanimity on the effects of radiation is due in part to insufficient data covering large population groups, from which agreed-on generalizations could be drawn.

For that is the one an increasingly large number of prominent Americans are now proposing.

In the first place, a large part of the discrepancy between President Eisenhower's estimate of a 1.5 billion dollar surplus for the same period and the new estimate of an almost seven billion dollar deficit is the result of the outgoing President's farewell gift of a political booby-trap to his successor.

Our complaint is that in many crucial areas the Kennedy programs are not too large but too small, most seriously in regard to the conventional arms build-up and in aid and welfare measures.

The resulting setup, it was declared, `` would be similar to that which is in successful operation in a number of metropolitan counties as large or larger than Rhode Island ''.

Another effect discovered is the large coefficient of thermal diffusion tending to separate nitrogen from the oxygen when temperature differences straddling the nitrogen dissociation region are present.

Mr. Speaker, for several years now the commuter railroads serving our large metropolitan areas have found it increasingly difficult to render the kind of service our expanding population wants and is entitled to have.