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Difference between revisions of "Residue"
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− | The [[Lucas-Lehmer test]], like [[division]], at the end produces a [[remainder]] of sorts, it is called the '''residue''' or residual. And like division, if the remainder/residue is zero there is a special (but different) meaning. For division a zero remaindert means that the number is exactly divisible. For the L-L test a zero residue means that the number is [[ | + | The [[Lucas-Lehmer test]], like [[division]], at the end produces a [[remainder]] of sorts, it is called the '''residue''' or residual. And like division, if the remainder/residue is zero there is a special (but different) meaning. For division a zero remaindert means that the number is exactly divisible. For the L-L test a zero residue means that the number is [[prime]]. |
The residue is the end result of the long calculation of the L-L test. When the numbers being tested are large: <math>\gt2^{64}-1</math> (i.e. exponents larger than 64) and above, the residue is 16 hexadecimal digits or longer. To ensure that no errors caused a [[Mersenne prime]] to be missed and that there cofidence in the results [[GIMPS]] is set-up to [[double check]] all L-L tests. To clear an [[exponent]] two matching residues must be reported. For two test to produced matching erroneaous residues (meaning they both missed a prime) out of a pool of ~ 18.4 pentillion numbers, this is considered to be impossible. | The residue is the end result of the long calculation of the L-L test. When the numbers being tested are large: <math>\gt2^{64}-1</math> (i.e. exponents larger than 64) and above, the residue is 16 hexadecimal digits or longer. To ensure that no errors caused a [[Mersenne prime]] to be missed and that there cofidence in the results [[GIMPS]] is set-up to [[double check]] all L-L tests. To clear an [[exponent]] two matching residues must be reported. For two test to produced matching erroneaous residues (meaning they both missed a prime) out of a pool of ~ 18.4 pentillion numbers, this is considered to be impossible. | ||
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==External links== | ==External links== | ||
− | *[ | + | *[[Wikipedia:Residue_number_system|Wikipedia]] |
[[Category:Math]] | [[Category:Math]] |
Latest revision as of 10:24, 6 February 2019
The Lucas-Lehmer test, like division, at the end produces a remainder of sorts, it is called the residue or residual. And like division, if the remainder/residue is zero there is a special (but different) meaning. For division a zero remaindert means that the number is exactly divisible. For the L-L test a zero residue means that the number is prime.
The residue is the end result of the long calculation of the L-L test. When the numbers being tested are large:
Mathematicians are interested in these residues. It is not practical for GIMPS to have the full length residues stored and sent to PrimeNet.
Example
Here is the Lucas test for
- S0 = 4
- S1 = (4 * 4 - 2) mod 127 = 14
- S2 = (14 * 14 - 2) mod 127 = 67
- S3 = (67 * 67 - 2) mod 127 = 42
- S4 = (42 * 42 - 2) mod 127 = 111
- S5 = (111 * 111 - 2) mod 127 = 0 <- the residue for this test; 127 is prime.