# Their Fundamental Nature

Prime Number

noun

a whole number that can only be divided without a remainder by itself and one.

Prime numbers are the building blocks of all numbers greater than 1. That is, every number is either itself a prime, such as 2,17,53 or 673, or is the product of primes, such as 17,119 (17 x 19 x 53). Furthermore, every number can be broken up into its primes in only one way. This is no mere supposition: in 1801 the mathematician Carl Gauss gave a proof of this “Fundamental Theorem of Arithmetic” (though it seems likely that Euclid had a proof of it 2,000 years earlier).

Beyond their fundamental nature, primes tantalize mathematicians with properties that seem to be true but defy proof. For example, Euclid himself came up with a wonderfully neat proof that there is an infinite number of primes, yet to this day no one has proved that there is an infinitude of “prime pairs”, such as 5 and 7 or 59 and 61, in which two consecutive odd numbers are primes.

Then there is Goldbach’s Conjecture, first pointed out in 1742, which states that every number greater than 5 is a unique sum of just three primes. Again, while this is widely believed to be true, no one has ever succeeded in proving it.

# Numbers, Games and Pastimes

Proving that a given number is prime has long been used to demonstrate calculating prowess. Once performed by “savants” with a gift for mental calculation, the task was among the first given to electronic computers. The current record for the largest known prime is 243,112,609– 1. It was found by electrical engineer Hans-Michael Elvenich on 6 Sept. 2008. It has 12,978,189 digits and was discovered using a network of thousands of linked home computers. (see our article To Surf and Serve)

Since the late 1970s, primes have become of enormous commercial importance, as they form the heart of the RSA encryption system, widely used to protect financial transactions.

Roughly speaking, the RSA system is based on the belief that there is no quick way to factorize big numbers into two similarly-sized prime numbers. While many think this is true, yet again a solid proof is lacking. Given what’s at stake, I find this rather disconcerting – equivalent to a bank declaring it’s pretty sure no one will find the mat under which it has put the keys to the safe.

All Prime Numbers up to 1,000
2 3 5 7 11 13 17 19 23
29 31 37 41 43 47 53 59 61 67
71 73 79 83 89 97 101 103 107 109
113 127 131 137 139 149 151 157 163 167
173 179 181 191 193 197 199 211 223 227
229 233 239 241 251 257 263 269 271 277
281 283 293 307 311 313 317 331 337 347
349 353 359 367 373 379 383 389 397 401
409 419 421 431 433 439 443 449 457 461
463 467 479 487 491 499 503 509 521 523
541 547 557 563 569 571 577 587 593 599
601 607 613 617 619 631 641 643 647 653
659 661 673 677 683 691 701 709 719 727
733 739 743 751 757 761 769 773 787 797
809 811 821 823 827 829 839 853 857 859
863 877 881 883 887 907 911 919 929 937
941 947 953 967 971 977 983 991 997

Author: Robert