Wieferich pair
Encyclopedia
In mathematics
, a Wieferich pair is a pair of prime number
s p and q that satisfy
Wieferich pairs are named after German
mathematician
Arthur Wieferich
.
There are only six Wieferich pairs known:
Wieferich pairs play an important role in Preda Mihăilescu
's 2002 proof of Mihăilescu's theorem
(formerly known as Catalan's conjecture).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Wieferich pair is a pair of prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
s p and q that satisfy
- pq − 1 ≡ 1 (modModular arithmeticIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
q2) and qp − 1 ≡ 1 (mod p2)
Wieferich pairs are named after German
Germany
Germany , officially the Federal Republic of Germany , is a federal parliamentary republic in Europe. The country consists of 16 states while the capital and largest city is Berlin. Germany covers an area of 357,021 km2 and has a largely temperate seasonal climate...
mathematician
Mathematician
A mathematician is a person whose primary area of study is the field of mathematics. Mathematicians are concerned with quantity, structure, space, and change....
Arthur Wieferich
Arthur Wieferich
Arthur Josef Alwin Wieferich was a German mathematician and teacher, remembered for his work on number theory....
.
There are only six Wieferich pairs known:
- (2, 1093), (3, 1006003), (5, 1645333507), (83, 4871), (911, 318917), and (2903, 18787) (sequence and in OEISOn-Line Encyclopedia of Integer SequencesThe On-Line Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
)
Wieferich pairs play an important role in Preda Mihăilescu
Preda Mihailescu
Preda V. Mihăilescu is a Romanian mathematician, best known for his proof of Catalan's conjecture.Born in Bucharest, he is the brother of Vintilă Mihăilescu. After leaving Romania in 1973, he settled in Switzerland. He studied mathematics and computer science in Zürich, receiving his Ph.D. from...
's 2002 proof of Mihăilescu's theorem
Mihailescu's theorem
Catalan's conjecture is a theorem in number theory that was conjectured by the mathematician Eugène Charles Catalan in 1844 and proven in 2002 by Preda Mihăilescu....
(formerly known as Catalan's conjecture).