There was a lot discussion on the net those last days concerning a big step toward the proof of a very old conjecture about prime numbers.
The conjecture states that there is an infinity of “twin prime numbers”, prime numbers separated only by two units, such as 11 and 13 for instance.
I was interested to the history of this conjecture and I tried some bibliographic research :
Some state that the conjecture was made by the Greek mathematician Euclid at 300 BC. Indeed Euclid had defined what a prime number is, and proved that there is an infinity of primes. (See the very elegant proof in Book 9 of the Elements, proposition 20, page 271 of this pdf). But I have not found the conjecture about twin primes in the Elements.
Most authors cite the French mathematician Alphonse de Polignac as the first to formulate the conjecture in a more general form at 1849 : in the Compte Rendus de l’Académie des Sciences, October 15th 1849, he published a paper on number theory, where he states among other theorems what has been later called the “de Polignac conjecture” :
“Every even number is equal to the difference of two prime numbers with an infinity of possibilities”.
If you consider the even number 2, you have the “twin primes” (like 11 and 13), with the number 4 you obtain what is called the “cousin primes” (like 13 and 17) and with the number 6 you have the so called “sexy primes” (like 11 and 17). And the infinity of possibilities implies that there is an infinity of each of these pairs, thus for n=2 we have the twin prime conjecture.
The “de Polignac conjecture” is a general case of the twin prime conjecture but is also a special case of the “Dickson conjecture” (formulated by the American mathematician Leonard Eugene Dickson at 1904), which covers also other special primes, such as the “Sophie Germain primes”, primes G such as 2*G+1 is also prime (2*G+1 is then called “safe prime”), named after the great French mathematician Marie-Sophie Germain. For instance 89 is a Sophie Germain prime because 2*89+1=179 is also prime, a safe prime.
Feel free to comment, especially if you find the proposition of Euclid about twin primes.
The conjecture states that there is an infinity of “twin prime numbers”, prime numbers separated only by two units, such as 11 and 13 for instance.
I was interested to the history of this conjecture and I tried some bibliographic research :
Some state that the conjecture was made by the Greek mathematician Euclid at 300 BC. Indeed Euclid had defined what a prime number is, and proved that there is an infinity of primes. (See the very elegant proof in Book 9 of the Elements, proposition 20, page 271 of this pdf). But I have not found the conjecture about twin primes in the Elements.
Most authors cite the French mathematician Alphonse de Polignac as the first to formulate the conjecture in a more general form at 1849 : in the Compte Rendus de l’Académie des Sciences, October 15th 1849, he published a paper on number theory, where he states among other theorems what has been later called the “de Polignac conjecture” :
“Every even number is equal to the difference of two prime numbers with an infinity of possibilities”.
If you consider the even number 2, you have the “twin primes” (like 11 and 13), with the number 4 you obtain what is called the “cousin primes” (like 13 and 17) and with the number 6 you have the so called “sexy primes” (like 11 and 17). And the infinity of possibilities implies that there is an infinity of each of these pairs, thus for n=2 we have the twin prime conjecture.
The “de Polignac conjecture” is a general case of the twin prime conjecture but is also a special case of the “Dickson conjecture” (formulated by the American mathematician Leonard Eugene Dickson at 1904), which covers also other special primes, such as the “Sophie Germain primes”, primes G such as 2*G+1 is also prime (2*G+1 is then called “safe prime”), named after the great French mathematician Marie-Sophie Germain. For instance 89 is a Sophie Germain prime because 2*89+1=179 is also prime, a safe prime.
Feel free to comment, especially if you find the proposition of Euclid about twin primes.
