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Enter (or paste) a positive decimal integer into the top text box (the one above the "Is N Prime", "Factor N", and "Is N Safe Prime" buttons). Then push one of the buttons to discover whether it is prime or Safe Prime, factor it, or find the next lower or higher prime or Safe Prime. Keep clicking "Lower" or "Higher" to find additional primes. You can also paste numbers directly into the Prime or Safe Prime text boxes.
Sophie Germain Primes
Sophie Germain Primes were named for Marie-Sophie Germain (1776-1831), the famous French woman mathematician, who first did research on them. A Sophie Germain Prime P is a prime for which 2P+1 is also prime.
A "Safe Prime" is a prime P for which (P-1)/2 is also prime. Note that this means that (P-1)/2 is a Sophie Germain Prime.
Safe Primes are important in cryptography because the modulus of a Diffie-Hellman Key Exchange MUST be a Safe Prime. The modulus of an RSA Public Key Pair MUST be the product of two primes, but some standards require that the modulus for an RSA key pair must be the product of a pair of Safe Primes.
The algorithm used here for factoring and prime testing is trial division by two, then by succeeding odd numbers: 3, 5, 7... Each divisor that is found is checked to see whether it might be a multiple factor. For example, 16 has four factors of two.
When factoring a number, it is only necessary to search factors up to its square root. This is because no number, by definition, can have two factors larger than its square root, and this algorithm finds the smallest ones first.
This is even easier for composite (non-prime) numbers. When a factor is found, the quotient remaining can be either composite or prime, but it cannot have any prime factors smaller than the most recent trial divisor (otherwise they would have already been found). So the same logic as to the largest possible remaining factors can be used on this new quotient, and the limit of the search for factors is reduced to the square root of the newly found quotient. This logic is used iteratively as more factors are found.
Eventually, the trial divisor is larger than the limit for whatever number is being factored at that point. This number is guaranteed to be prime and is output as the last factor.
If the original Dividend were really prime, the limit would be its square root, and would never be reduced, which is why with this particular algorithm it takes a lot longer to prove that a prime number is really prime than to factor a composite number. Primality tests other than trying successive trial divisors do exist, and are stupendously faster when large numbers are involved. For extremely large numbers (hundreds or thousands of digits) it is far easier to find out whether a number is prime than it would be to factor it. That reality is part of the mathematical basis of the "trap door" function used in the RSA system of public key cryptography.
The "Is This Prime?" button is provided because most large composite numbers have at least one small prime factor, and stopping after finding the first factor (at which point you know the number is not prime) can take a lot less time than continuing on to find the large factors.
The simplified function connected with the "Is This Prime?" button is also used by the "Next Lower Prime" and "Next Higher Prime" buttons. The algorithm here begins by checking if the number in the box is even or odd. If even, it starts its search one above or below where it is. If the number is odd, it starts two above or below. Once started, it keeps stepping in the chosen direction by two until it finds a prime. Naturally, this takes a while because you have to search a fairly large number of candidates to find another prime.
Searching for "Safe Primes" requires finding a prime P, then testing whether (P-1)/2 is prime, and moving on if it isn't. Thus finding Safe Primes is much more compute intensive than finding regular primes.
Some academic and government work (some of it secret, apparently) is going on in the area of factoring composite numbers consisting of the product of two large primes. If you could do this easily, you could invalidate the security of cryptographic systems that use the RSA system of Public Keys. Examples include secure web site certificates, and most forms of encrypted EMAIL. Some groups (e.g., the CIA and the Pentagon) would like to have this ability to do this at least for "emergency" purposes, and this apparently fuels research expenditures. Unfortunately for these efforts, factoring large composites is known to be an "NP complete" problem, which means that if you could do this easily, at least in principle, you could solve any mathematical problem easily. Quantum computing offers the possibility of doing this essentially by "brute force".
Another, entirely different, approach to factorization is use of " Quantum Computers".
The table to the left gives times required to attempt to factor some large primes on my computer, which is a 2001 model 600 MHz G3 iMac running Mac OSX 10.3.4. Times are given in Minutes:Seconds.
Browsers used for this test were:
My only reward for writing this is the 15 milliseconds of fame I receive from having my name here. Don't deprive me of that.
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