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A prime number is an integer, or whole number, that has only two factors — 1 and itself. Put another way, a prime number can be divided evenly only by 1 and by itself. Prime numbers also must be greater than 1. For example, 3 is a prime number, because 3 cannot be divided evenly by any number except for 1 and 3. However, 6 is not a prime number, because it can be divided evenly by 2 or 3.
List of prime numbers
The prime numbers between 1 and 1,000 are:
Swipe to scroll horizontally
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
Row 18 - Cell 6
Row 18 - Cell 7
Row 18 - Cell 8
Largest prime number
The largest prime number discovered so far is 2 raised to the 57,885,161st power minus 1, or 257,885,161 - 1. It is 17,425,170 digits long. It was discovered by University of Central Missouri mathematician Curtis Cooper as part of a giant network of volunteer computers devoted to finding primes.
Prime numbers have been studied for thousands of years. Euclid's "Elements," published about 300 B.C., proved several results about prime numbers. In Book IX of the "Elements," Euclid writes that there are infinitely many prime numbers. Euclid also provides proof of the Fundamental Theorem of Arithmetic — every integer can be written as a product of primes in a unique way. In "Elements," Euclid solves the problem of how to create a perfect number, which is a positive integer equal to the sum of its positive divisors, using Mersenne primes. A Mersenne prime is a prime number that can be calculated with the equation 2n-1. [Countdown: The Most Massive Numbers in Existence]
This grid can be used as a Sieve of Eratosthenes if you were to cross out all of the numbers that are multiples of other numbers. The prime numbers are underlined.
In 200 B.C., Eratosthenes created an algorithm that calculated prime numbers, known as the Sieve of Eratosthenes. This algorithm is one of the earliest algorithms ever written. Eratosthenes put numbers in a grid, and then crossed out all multiples of numbers until the square root of the largest number in the grid is crossed out. For example, with a grid of 1 to 100, you would cross out the multiples of 2, 3, 4, 5, 6, 7, 8, 9, and 10, since 10 is the square root of 100. Since 6, 8, 9 and 10 are multiples of other numbers, you no longer need to worry about those multiples. So for this chart, you would cross out the multiples of 2, 3, 5 and 7. With these multiples crossed out, the only numbers that remain and are not crossed out are prime. This sieve enables someone to come up with large quantities of prime numbers.
But during the Dark Ages, when intellect and science were suppressed, no further work was done with prime numbers. In the 17th century, mathematicians like Fermat, Euler and Gauss began to examine the patterns that exist within prime numbers. The conjectures and theories put out by mathematicians at the time revolutionized math, and some have yet to be proven to this day. In fact, proof of the Riemann Hypothesis, based on Bernhard Riemann's theory about patterns in prime numbers, carries a $1 million prize from the Clay Mathematics Institute. [Related: Famous Prime Number Conjecture One Step Closer to Proof]
Prime numbers & encryption
In 1978, three researchers discovered a way to scramble and unscramble coded messages using prime numbers. This early form of encryption paved the way for Internet security, putting prime numbers at the heart of electronic commerce. Public-key cryptography, or RSA encryption, has simplified secure transactions of all times. The security of this type of cryptography relies on the difficulty of factoring large composite numbers, which is the product of two large prime numbers.
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Confidence in modern banking and commerce systems hinges on the assumption that large composite numbers cannot be factored in a short amount of time. Two primes are considered as sufficiently secure if they are 2,048 bits long, because the product of these two primes would be about 1,234 decimal digits.
Prime numbers in nature
Prime numbers even show up in nature. Cicadas spend most of their time hiding, only reappearing to mate every 13 or 17 years. Why this specific number? Scientists theorize that cicadas reproduce in cycles that minimize possible interactions with predators. Any predator reproductive cycle that divides the cicada's cycle evenly means that the predator will hatch out the same time as the cicada at some point. For example, if the cicada evolved towards a 12-year reproductive cycle, predators who reproduce at the 2, 3, 4 and 6 year intervals would find themselves with plenty of cicadas to eat. By using a reproductive cycle with a prime number of years, cicadas would be able to minimize contact with predators.
This may sound implausible (obviously, cicadas don't know math), but simulation models of 1,000 years of cicada evolution prove that there is a major advantage for reproductive cycle times based on primes. It can be viewed here at http://www.arachnoid.com/prime_numbers/. It may not be intentional on the part of Mother Nature, but prime numbers show up more in nature and our surrounding world than we may think.
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