François Proth (22 March 1852 – 21 January 1879) was a French self-taught mathematician farmer who lived in
Vaux-devant-Damloup
Vaux-devant-Damloup (, literally ''Vaux before Damloup'') is a former commune in the Meuse department in Grand Est in north-eastern France. It had a population of 76 (2019). On 1 January 2019, it was merged into the new commune Douaumont-Vaux.< ...
near
Verdun
Verdun (, , , ; official name before 1970 ''Verdun-sur-Meuse'') is a large city in the Meuse department in Grand Est, northeastern France. It is an arrondissement of the department.
Verdun is the biggest city in Meuse, although the capital ...
,
France
France (), officially the French Republic ( ), is a country primarily located in Western Europe. It also comprises of Overseas France, overseas regions and territories in the Americas and the Atlantic Ocean, Atlantic, Pacific Ocean, Pac ...
.
He stated four primality-related theorems. The most famous of these,
Proth's theorem In number theory, Proth's theorem is a primality test for Proth numbers.
It states that if ''p'' is a Proth number, of the form ''k''2''n'' + 1 with ''k'' odd and ''k'' < 2''n'', and if there exists an
, can be used to test whether a
Proth number
A Proth number is a natural number ''N'' of the form N = k \times 2^n +1 where ''k'' and ''n'' are positive integers, ''k'' is odd and 2^n > k. A Proth prime is a Proth number that is prime. They are named after the French mathematician François ...
(a number of the form ''k''2''n'' + 1 with ''k'' odd and ''k'' < 2''n'') is prime. The numbers passing this test are called
Proth prime
A Proth number is a natural number ''N'' of the form N = k \times 2^n +1 where ''k'' and ''n'' are positive integers, ''k'' is odd and 2^n > k. A Proth prime is a Proth number that is prime. They are named after the French mathematician François ...
s; they continue to be of importance in the computational search for large prime numbers.
Proth also formulated
Gilbreath's conjecture
Gilbreath's conjecture is a conjecture in number theory regarding the sequences generated by applying the forward difference operator to consecutive prime numbers and leaving the results unsigned, and then repeating this process on consecutive ter ...
on successive differences of primes, 80 years prior to Gilbreath, but his proof of the conjecture turned out to be erroneous..
The cause of Proth's death is not known.