In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy
Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof [1] of Mihăilescu's theorem (formerly known as Catalan's conjecture). [2]
There are only 7 Wieferich pairs known: [3] [4]
A Wieferich triple is a triple of prime numbers p, q and r that satisfy
There are 17 known Wieferich triples:
Barker sequence or Wieferich n-tuple is a generalization of Wieferich pair and Wieferich triple. It is primes (p1, p2, p3, ..., pn) such that
For example, (3, 11, 71, 331, 359) is a Barker sequence, or a Wieferich 5-tuple; (5, 188748146801, 453029, 53, 97, 76704103313, 4794006457, 12197, 3049, 41) is a Barker sequence, or a Wieferich 10-tuple.
For the smallest Wieferich n-tuple, see OEIS: A271100, for the ordered set of all Wieferich tuples, see OEIS: A317721.
Wieferich sequence is a special type of Barker sequence. Every integer k>1 has its own Wieferich sequence. To make a Wieferich sequence of an integer k>1, start with a(1)=k, a(n) = the smallest prime p such that a(n-1)p-1 = 1 (mod p) but a(n-1) ≠ 1 or -1 (mod p). It is a conjecture that every integer k>1 has a periodic Wieferich sequence. For example, the Wieferich sequence of 2:
The Wieferich sequence of 83:
The Wieferich sequence of 59: (this sequence needs more terms to be periodic)
However, there are many values of a(1) with unknown status. For example, the Wieferich sequence of 3:
The Wieferich sequence of 14:
The Wieferich sequence of 39:
It is unknown that values for k exist such that the Wieferich sequence of k does not become periodic. Eventually, it is unknown that values for k exist such that the Wieferich sequence of k is finite.
When a(n - 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown)
In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy
Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof [1] of Mihăilescu's theorem (formerly known as Catalan's conjecture). [2]
There are only 7 Wieferich pairs known: [3] [4]
A Wieferich triple is a triple of prime numbers p, q and r that satisfy
There are 17 known Wieferich triples:
Barker sequence or Wieferich n-tuple is a generalization of Wieferich pair and Wieferich triple. It is primes (p1, p2, p3, ..., pn) such that
For example, (3, 11, 71, 331, 359) is a Barker sequence, or a Wieferich 5-tuple; (5, 188748146801, 453029, 53, 97, 76704103313, 4794006457, 12197, 3049, 41) is a Barker sequence, or a Wieferich 10-tuple.
For the smallest Wieferich n-tuple, see OEIS: A271100, for the ordered set of all Wieferich tuples, see OEIS: A317721.
Wieferich sequence is a special type of Barker sequence. Every integer k>1 has its own Wieferich sequence. To make a Wieferich sequence of an integer k>1, start with a(1)=k, a(n) = the smallest prime p such that a(n-1)p-1 = 1 (mod p) but a(n-1) ≠ 1 or -1 (mod p). It is a conjecture that every integer k>1 has a periodic Wieferich sequence. For example, the Wieferich sequence of 2:
The Wieferich sequence of 83:
The Wieferich sequence of 59: (this sequence needs more terms to be periodic)
However, there are many values of a(1) with unknown status. For example, the Wieferich sequence of 3:
The Wieferich sequence of 14:
The Wieferich sequence of 39:
It is unknown that values for k exist such that the Wieferich sequence of k does not become periodic. Eventually, it is unknown that values for k exist such that the Wieferich sequence of k is finite.
When a(n - 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown)