Pseudoprime




A pseudoprime is a probable prime (an integer that shares a property common to all prime numbers) that is not actually prime. Pseudoprimes are classified according to which property of primes they satisfy.


Some sources use the term pseudoprime to describe all probable primes, both composite numbers and actual primes.


Pseudoprimes are of primary importance in public-key cryptography, which makes use of the difficulty of factoring large numbers into their prime factors. Carl Pomerance estimated in 1988 that it would cost $10 million to factor a number with 144 digits, and $100 billion to factor a 200-digit number (the cost today is dramatically cheaper but still prohibitively expensive).[1][2] However, finding and factoring the proper prime numbers for this use is correspondingly expensive, so various probabilistic primality tests are used to find primes among large numbers, some of which in rare cases incorrectly identify composite numbers as primes. On the other hand, deterministic primality tests, such as the AKS primality test, do not give false positives; there are no pseudoprimes with respect to them.



Fermat pseudoprimes


Main article: Fermat pseudoprime

Fermat's little theorem states that if p is prime and a is coprime to p, then ap−1 − 1 is divisible by p. For an integer a > 1, if a composite integer x divides ax−1 − 1, then x is called a Fermat pseudoprime to base a. It follows that if x is a Fermat pseudoprime to base a, then x is coprime to a. Some sources use variations of this definition, for example to allow only odd numbers to be pseudoprimes.[3]


An integer x that is a Fermat pseudoprime to all values of a that are coprime to x is called a Carmichael number.



Classes




  • Catalan pseudoprime

  • Elliptic pseudoprime

  • Euler pseudoprime

  • Euler–Jacobi pseudoprime

  • Fermat pseudoprime

  • Frobenius pseudoprime

  • Lucas pseudoprime

  • Perrin pseudoprime

  • Somer–Lucas pseudoprime

  • Strong pseudoprime




References





  1. ^ Clawson, Calvin C. (1996). Mathematical Mysteries: The Beauty and Magic of Numbers. Cambridge: Perseus. p. 195. ISBN 0-7382-0259-2..mw-parser-output cite.citation{font-style:inherit}.mw-parser-output q{quotes:"""""""'""'"}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-limited a,.mw-parser-output .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}


  2. ^ Cipra, Barry A. (December 23, 1988). "PCs Factor a 'Most Wanted' Number". Science. 242: 1634–1635. doi:10.1126/science.242.4886.1634. PMID 17730568.


  3. ^ Weisstein, Eric W. "Fermat Pseudoprime". MathWorld.











-

Popular posts from this blog

Bressuire

Image
Subprefecture and commune in Nouvelle-Aquitaine, France Bressuire Subprefecture and commune Chateau de Bressuire and the Eglise Notre-Dame Coat of arms Location of Bressuire Bressuire Show map of France Bressuire Show map of Nouvelle-Aquitaine Coordinates: 46°50′27″N 0°29′14″W  /  46.8408°N 0.4872°W  / 46.8408; -0.4872 Coordinates: 46°50′27″N 0°29′14″W  /  46.8408°N 0.4872°W  / 46.8408; -0.4872 Country France Region Nouvelle-Aquitaine Department Deux-Sèvres Arrondissement Bressuire Canton Bressuire Government  • Mayor .mw-parser-output .nobold{font-weight:normal} (2014–20) Jean Michel Bernier Area 1 180.59 km 2 (69.73 sq mi) Population (2014) 2 19,300  • Density 110/km 2 (280/sq mi) Time zone UTC+01:00 (CET)  • Summer (DST) UTC+02:00 (CEST) INSEE/Postal code 79049 /79300 Elevation 98–236 m (322–774 ft) (avg. 173 m or 568 ft) 1 French Land Register data, which exclude
Read more

Vorschmack

Image
Vorschmack Ukrainian Jewish-style vorschmack served on rye bread Course Hors d'oeuvre Region or state Eastern Europe Associated national cuisine Ashkenazi Jewish, Finnish, German, Ukrainian, Polish, Russian Main ingredients Ground meat and/or fish Cookbook: Vorschmack   Media: Vorschmack Vorschmack or forshmak (Yiddish: פֿאָרשמאַק ‎, from archaic German Vorschmack , "foretaste" [1] or "appetizer" [2] ) is an originally East European dish made of salty minced fish or meat. Different variants of this dish are especially common in Ashkenazi Jewish and Finnish cuisine. Some varieties are also known in Russian and Polish cuisine. Contents 1 In Jewish cuisine 2 In Russian cuisine 3 In Polish cuisine 4 In Finnish cuisine 5 See also 6 References In Jewish cuisine According to Gil Marks, the German name points to the possible Germanic origin of this dish. [1] William Pokhlyobkin descr
Read more

Quarantine

Image
For other uses, see Quarantine (disambiguation). Signal flag "Lima" called the "Yellow Jack" which when flown in harbor means ship is under quarantine. A simple yellow flag (also called the "Yellow Jack") had historically been used to signal quarantine (it stands for Q among signal flags), but now indicates the opposite, as a signal of a ship free of disease that requests boarding and inspection. A quarantine is used to separate and restrict the movement of people; it is 'a restraint upon the activities or communication of persons or the transport of goods designed to prevent the spread of disease or pests', for a certain period of time. [1] This is often used in connection to disease and illness, such as those who may possibly have been exposed to a communicable disease. [2] The term is often erroneously used to mean medical isolation, which is "to separate ill persons who have a communicable disease from those who are healthy
Read more