Wieferich Prime Pairs, Barker Sequences,
and Circulant Hadamard Matrices

This page contains data associated with my article Wieferich pairs and Barker sequences, including lists of Wieferich prime pairs found in computations for the article, and integers n that have not been eliminated as the possible length of a long Barker sequence, or as the order of a large circulant Hadamard matrix.

Recall that a Wieferich prime pair (q, p) has the property that q^p-1 = 1 mod p². A Barker sequence is a finite sequence {a_i}, each term ±1, for which each sum of the form ∑_i a_ia_i+k with k ≠ 0 is -1, 0, or 1. A circulant Hadamard matrix of order n is an n × n matrix of ±1's whose rows are mutually orthogonal, and each of whose rows after the first is obtained from the prior one by cyclically shifting its elements by one position to the right.

1. Wieferich Prime Pairs: Unrestricted searches. (In each list except the last, we only show pairs with q < p.)
   • q = 3 or 5 and p < 10¹⁴; q = 7 and 7 < p < 10¹³ (9 pairs): q1p14.txt.
   • 10 < q < 100, q < p < 10¹² (34 pairs): q2p12.txt. Also includes the (empty) results of testing q = 17 with 10¹² < p < 10¹⁴.
   • 100 < q < 1000, q < p < 10¹¹ (196 pairs): q3p11.txt.
   • 1000 < q < 10⁴, q < p < 10¹⁰ (1108 pairs): q4p10.txt.
   • 10⁴ < q < 10⁵, q < p < 10⁹ (5486 pairs): q5p9.txt.
   • 10⁵ < q < 10⁶, q < p < 10⁸ (23645 pairs): q6p8.txt.
   • 10⁶ < q < p < 10⁷ (29359 pairs): q7p7.txt.
   • Combined list with all of the above pairs (59837 pairs): all_wiefpairs_asc.txt.gz.
   • All pairs with p and q odd and max{p, q} < 10⁷ (1689589 pairs): all_wiefpairs_10m.txt.gz.


2. Wieferich Prime Pairs: Additional pairs found where both p and q are 1 mod 4, again restricting to pairs with q < p.
   • No additional pairs found with q < 100 and 10¹² < p < 10¹³.
   • 100 < q < 1000, 10¹¹ < p < 10¹² (3 pairs): extra_q3.txt.
   • 1000 < q < 10⁴, 10¹⁰ < p < 10¹¹ (20 pairs): extra_q4.txt.
   • 10⁴ < q < 10⁵, 10⁹ < p < 10¹⁰ (240 pairs): extra_q5.txt.
   • 10⁵ < q < 10⁶, 10⁸ < p < 7.5 · 10⁹ (3728 pairs): extra_q6.txt.
   • 10⁶ < q < 10⁷, 10⁷ < p < 7.5 · 10⁸ (34849 pairs): extra_q7.txt.
   • 10⁷ < q < p < 10⁸ (54323 pairs): extra_q8.txt.
   • Combined list with all of the above pairs (93163 pairs): extra_wiefpairs_asc.txt.gz.


3. Higher-Order Wieferich Prime Pairs with p > 3: q^p-1 = 1 mod p^k and k > 2.
   • k = 3: wieferich3.txt (53116 pairs).
   • k = 4: wieferich4.txt (7159 pairs).
   • k = 5: wieferich5.txt (1170 pairs).
   • k = 6: wieferich6.txt (215 pairs).
   • k = 7: wieferich7.txt (39 pairs).
   • k = 8: wieferich8.txt (7 pairs).
   • k = 9: wieferich9.txt (1 pair).


4. Permissible Lengths of Barker Sequences.
   • Only one value of n ≤ 2 · 10³⁰ has not been eliminated as the possible length of a Barker sequence: n = 189260468001034441522766781604. Here, n = 4m² with m = 217520382953549 = 13 · 41 · 2953 · 138200401.


5. Permissible Orders of Circulant Hadamard Matrices.
   • The 1578 values of m ≤ 10¹³ for which n = 4m² has not been eliminated as the possible order of a circulant Hadamard matrix.

Related Links

• Additional data and updated results associated with Wieferich primes and Barker sequences, from the second article on this topic by P. Borwein and M. J. Mossinghoff.
• Fermat quotients q_p(a) that are divisible by p, by Wilfrid Keller and Jörg Richstein.
• Fermat quotients, by Richard Fischer.

Last modified May 31, 2013.