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Prime Numbers - the Search and the Discovery

The main mathematical news

1976: The use of prime numbers in cryptography.

2008: The discovery of the first prime number with more than 10 million digits.

2017: The discovery of the 50th Mersenne prime with more than 23 million digits.

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Additional Theorems / conjectures / Open questions

*  There are infinitely many prime numbers.

*  Mersenne’s (refuted) conjecture:  is prime for p=2,3,5,7,13,17,19,31,67,127,257

*  Is the set of Mersenne primes infinite?

*  Many Mersenne numbers are composite.

*  Many primes aren’t Mersenne numbers.

*  For all natural numbers n between 1 and 40, the number M of the form  is prime. Is there a “formula” that generates all and only prime numbers?

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The main mathematical concepts / Principles

Number theory (MSC2010#97F60) 

*    Prime/Composite (natural) number

*    Mersenne prime

Logic (MSC2010#97E30) 

*    Proof by contradiction

Real life mathematics (MSC2010#97F90) 

*    Cryptography

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