- Semiprime
In

mathematics , a**semiprime**(also called**biprime**or**2-**, oralmost prime **pq number**) is anatural number that is the product of two (not necessarily distinct)prime number s. The first few semiprimes are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, ... OEIS|id=A001358.As of|2008|September, the largest known semiprime is (2

^{43,112,609}− 1)^{2}, which has over 25 million digits. This is the square of thelargest known prime . The square of any prime number is a semiprime, so the largest known semiprime will always be the square of the largest known prime, unless the factors of the semiprime are not known. It is conceivable that somebody could find a way to prove a larger number is a semiprime without knowing the two factors, but so far that has only happened for smaller semiprimes. [*Chris Caldwell, [*]*http://primes.utm.edu/glossary/page.php?sort=Semiprime "The Prime Glossary: semiprime"*] at ThePrime Pages . Retrieved on2007 -12-04.The value of

Euler's totient function for a semiprime "n" = "pq" is particularly simple when "p" and "q" are distinct::φ("n") = "n" + 1 − ("p" + "q").

**Applications**Semiprimes are highly useful in the area of

cryptography andnumber theory , most notably inpublic key cryptography , where they are used byRSA andpseudorandom number generator s such asBlum Blum Shub . These methods rely on the fact that finding two large primes and multiplying them together is computationally simple, whereas finding the original factors appears to be difficult. In theRSA Factoring Challenge ,RSA Security offered prizes for the factoring of specific large semiprimes and several prizes were awarded. The most recent such challenge closed in 2007. [*[*]*http://www.rsa.com/rsalabs/node.asp?id=2092*]In practical cryptography, it is not sufficient to choose just any semiprime; a good number must evade a number of well-known special-purpose algorithms that can factor numbers of certain form. The factors "p" and "q" of "n" should be very large, around the same order of magnitude as the square root; this makes

trial division andPollard's rho algorithm impractical. At the same time they cannot be too close together, or else another simple test can factor the number. The number may also be chosen so that none of "p" − 1, "p" + 1, "q" − 1, or "q" + 1 aresmooth number s, protecting against Pollard's "p" − 1 algorithm or Williams' "p" + 1 algorithm. These checks cannot take future algorithms or secret algorithms into account however, introducing the possibility that numbers in use today may be broken by special-purpose algorithms.In 1974 the

Arecibo message was sent with a radio signal aimed at astar cluster . It consisted of 1679 binary digits intended to be interpreted as a 23×73bitmap image. The number 1679 = 23×73 was chosen because it is a semiprime and therefore can only be broken down into 23 rows and 73 columns, or 73 rows and 23 columns.**ee also***

Chen's theorem **References****External links***

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**semiprime**— noun A natural number that is the product of two prime numbers … Wiktionary**Semiprime ring**— In mathematics, a ring R is a semiprime ring if its zero ideal is the intersection of all its prime ideals.External links* [http://planetmath.org/encyclopedia/SemiprimeIdeal.html PlanetMath article on semiprime ideals] … Wikipedia**14 (number)**— ← 13 15 → 14 ← 10 11 12 13 14 15 16 … Wikipedia**15 (number)**— This article is about the number 15. For the year, see 15. For other uses, see 15 (disambiguation). ← 14 16 → 15 ← … Wikipedia**Ideal (ring theory)**— In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like even number or multiple of 3 . For instance, in… … Wikipedia**21 (number)**— For other uses, see 21 (disambiguation). ← 20 22 → 21 ← 20 21 22 … Wikipedia**33 (number)**— ← 32 34 → 33 ← 30 31 32 33 34 35 36 … Wikipedia**22 (number)**— This article is about the number. For other uses, see 22 (disambiguation). ← 21 23 → 22 ← 20 21 22 … Wikipedia**38 (number)**— This article discusses the number thirty eight. For the year 38 CE, see 38. For other uses of 38, see 38 (disambiguation) ← 37 39 → 38 ← … Wikipedia**46 (number)**— ← 45 47 → 46 ← 40 41 42 43 44 45 46 … Wikipedia