Liczby pierwsze Sophie Germain

Marie-Sophie Germain was born in Paris on 1 April 1776 to a prosperous silk merchant who sat in the Estates-General. She was thirteen when the Bastille fell, and the standard account has her retreating into her father's library during the unrest and finding mathematics in a history by Jean-Étienne Montucla, including its account of Archimedes killed while absorbed in a diagram. The further details often attached to the story — parents confiscating her candles, ink frozen in the inkwell, the girl discovered asleep over her slate — are repeated everywhere but rest on thin evidence and are best treated as embellishment.

The École Polytechnique opened in 1794 and did not admit women. Germain obtained the lecture notes and submitted coursework to Joseph-Louis Lagrange under the name Antoine-Auguste Le Blanc, a real student who had left the school. Lagrange was impressed enough to seek out the author and became a mentor when he discovered who she was. She used the same pseudonym from 1804 in a correspondence with Carl Friedrich Gauss about the Disquisitiones Arithmeticae. In 1807, with French troops occupying Gauss's Braunschweig and Archimedes presumably on her mind, she asked a family friend, General Joseph-Marie Pernety, to check on his safety. The intervention revealed her identity; Gauss wrote back with evident delight.

Her number theory produced the result that carries her name: if p and 2p + 1 are both prime, then any solution of x^p + y^p = z^p would need p to divide xyz. Adrien-Marie Legendre published it, with credit, in a footnote to his 1823 memoir to the Académie, and it stood as the strongest general progress on Fermat's Last Theorem for decades. Modern study of her surviving manuscripts shows the footnote was a fragment of a far more ambitious programme. She also won the Académie's prize for her work on vibrating elastic surfaces in 1816, at the third attempt, the first woman to take one. She died of breast cancer in 1831; her death certificate recorded her occupation as property holder, not mathematician.

Najważniejsze właściwości

• p is a Sophie Germain prime when p and 2p + 1 are both prime; 2p + 1 is then called a safe prime.
• The sequence begins 2, 3, 5, 11, 23, 29, 41, 53, 83, 89, 113, 131, 173, 179, 191.
• Apart from p = 3, every Sophie Germain prime is congruent to 2 modulo 3 — otherwise 2p + 1 would be a multiple of 3.
• Apart from 2, 3 and 5, every Sophie Germain prime ends in 1, 3 or 9: a prime ending in 7 would give a 2p + 1 ending in 5.
• If p ≡ 3 (mod 4) is a Sophie Germain prime then 2p + 1 divides the Mersenne number 2^p - 1, making it composite — this is why 23 divides 2047.
• Germain's theorem: if p and 2p + 1 are both prime, then x^p + y^p = z^p has no integer solution in which p fails to divide xyz.
• A Cunningham chain of the first kind is a run p, 2p + 1, 4p + 3, … of primes; 2, 5, 11, 23, 47 is one of length five.
• Whether infinitely many exist is unproved. The Hardy–Littlewood heuristic predicts about 1.32·x/(ln x)² of them below x, where the constant is twice the twin prime constant.

Gdzie się pojawiają

• Diffie–Hellman key exchange uses safe primes so that the multiplicative group has no small subgroups for an attacker to work in; the standard MODP groups published in RFC 3526 are all safe primes.
• OpenSSL's dhparam utility generates safe primes by default, searching for a prime p whose (p-1)/2 is also prime.
• The Blum Blum Shub pseudorandom generator is recommended to be built from safe primes, which maximise the cycle length of its squaring map.
• PrimeGrid runs a dedicated volunteer Sophie Germain Prime Search; the records now run to hundreds of thousands of digits.
• Germain primes were the tool behind the "first case" of Fermat's Last Theorem, which is where they entered number theory and where the name comes from — she did not name them herself.
• Germain is not among the seventy-two scientists whose names are inscribed on the Eiffel Tower, an omission frequently noted in writing about her.