Wall–Sun–Sun prime

In number theory, a Wall–Sun–Sun prime or Fibonacci–Wieferich prime is a certain kind of prime number which is conjectured to exist, although none are known.

Definition

Let <math>p</math> be a prime number. When each term in the sequence of Fibonacci numbers <math>F_n</math> is reduced modulo <math>p</math>, the result is a periodic sequence. The (minimal) period length of this sequence is called the Pisano period and denoted <math>\pi(p)</math>. Since <math>F_0 = 0</math>, it follows that p divides <math>F_{\pi(p)}</math>. A prime p such that p² divides <math>F_{\pi(p)}</math> is called a Wall–Sun–Sun prime.

Equivalent definitions

If <math>\alpha(m)</math> denotes the rank of apparition modulo <math>m</math> (i.e., <math>\alpha(m)</math> is the smallest positive index such that <math>m</math> divides <math>F_{\alpha(m)}</math>), then a Wall–Sun–Sun prime can be equivalently defined as a prime <math>p</math> such that <math>p^2</math> divides <math>F_{\alpha(p)}</math>.

For a prime p ≠ 2, 5, the rank of apparition <math>\alpha(p)</math> is known to divide <math>p - \left(\tfrac{p}{5}\right)</math>, where the Legendre symbol <math>\textstyle\left(\frac{p}{5}\right)</math> has the values

<math>\left(\frac{p}{5}\right) = \begin{cases} 1 &\text{if }p \equiv \pm1 \pmod 5;\\ -1 &\text{if }p \equiv \pm2 \pmod 5.\end{cases}</math>

This observation gives rise to an equivalent characterization of Wall–Sun–Sun primes as primes <math>p</math> such that <math>p^2</math> divides the Fibonacci number <math>F_{p - \left(\frac{p}{5}\right)}</math>.<ref name=Elsenhans2010>Template:Cite arXiv</ref>

A prime <math>p</math> is a Wall–Sun–Sun prime if and only if <math>\pi(p^2) = \pi(p)</math>.

A prime <math>p</math> is a Wall–Sun–Sun prime if and only if <math>L_p \equiv 1 \pmod{p^2}</math>, where <math>L_p</math> is the <math>p</math>-th Lucas number.<ref>Template:Cite journal</ref>Template:Rp

McIntosh and Roettger establish several equivalent characterizations of Lucas–Wieferich primes.<ref name="McIntosh Roettger"/> In particular, let <math>\epsilon = \left(\tfrac{p}{5}\right)</math>; then the following are equivalent:

• <math>F_{p - \epsilon} \equiv 0 \pmod{p^2}</math>
• <math>L_{p - \epsilon} \equiv 2\epsilon \pmod{p^4}</math>
• <math>L_{p - \epsilon} \equiv 2\epsilon \pmod{p^3}</math>
• <math>F_p \equiv \epsilon \pmod{p^2}</math>
• <math>L_p \equiv 1 \pmod{p^2}</math>

Existence

In a study of the Pisano period <math>k(p)</math>, Donald Dines Wall determined that there are no Wall–Sun–Sun primes less than <math>10000</math>. In 1960, he wrote:<ref name=Wall1960/> Template:Quote It has since been conjectured that there are infinitely many Wall–Sun–Sun primes.<ref>Template:Citation.</ref>

In 2007, Richard J. McIntosh and Eric L. Roettger showed that if any exist, they must be > 2Template:E.<ref name="McIntosh Roettger">Template:Cite journal</ref> Dorais and Klyve extended this range to 9.7Template:E without finding such a prime.<ref>{{#invoke:citation/CS1|citation |CitationClass=web }}</ref>

In December 2011, another search was started by the PrimeGrid project;<ref>Wall–Sun–Sun Prime Search project at PrimeGrid</ref> however, it was suspended in May 2017.<ref>[1] at PrimeGrid</ref> In November 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously.<ref>Message boards : Wieferich and Wall-Sun-Sun Prime Search at PrimeGrid</ref> The project ended in December 2022, proving that any Wall–Sun–Sun prime must exceed <math>2^{64}</math> (about <math>18\cdot 10^{18}</math>).<ref>Subproject status at PrimeGrid</ref>

History

Wall–Sun–Sun primes are named after Donald Dines Wall,<ref name=Wall1960>Template:Citation</ref><ref>Template:Cite journal</ref> Zhi Hong Sun and Zhi Wei Sun; Z. H. Sun and Z. W. Sun showed in 1992 that if the first case of Fermat's Last Theorem was false for a certain prime p, then p would have to be a Wall–Sun–Sun prime.<ref>Template:Citation</ref> As a result, prior to Wiles's proof of Fermat's Last Theorem, the search for Wall–Sun–Sun primes was also the search for a potential counterexample to this centuries-old conjecture.

Generalizations

A tribonacci–Wieferich prime is a prime p satisfying Template:Nowrap, where h(m) is the least positive integer k satisfying [T_k,T_k+1,T_k+2] ≡ [T₀, T₁, T₂] (mod m) and T_n denotes the n-th tribonacci number. No tribonacci–Wieferich prime exists below 10¹¹.<ref>Template:Cite journal</ref>

A Pell–Wieferich prime is a prime p satisfying p² divides P_p−1, when p congruent to 1 or 7 (mod 8), or p² divides P_p+1, when p congruent to 3 or 5 (mod 8), where P_n denotes the n-th Pell number. For example, 13, 31, and 1546463 are Pell–Wieferich primes, and no others below 10⁹ (sequence A238736 in the OEIS). In fact, Pell–Wieferich primes are 2-Wall–Sun–Sun primes.

Near-Wall–Sun–Sun primes

A prime p such that <math>F_{p - \left(\frac{p}{5}\right)} \equiv Ap \pmod{p^2}</math> with small |A| is called near-Wall–Sun–Sun prime.<ref name="McIntosh Roettger"/> Near-Wall–Sun–Sun primes with A = 0 would be Wall–Sun–Sun primes. PrimeGrid recorded cases with |A| ≤ 1000.<ref>Reginald McLean and PrimeGrid, WW Statistics</ref> A dozen cases are known where A = ±1 (sequence A347565 in the OEIS).

Wall–Sun–Sun primes with discriminant D

Wall–Sun–Sun primes can be considered for the field <math>Q_{\sqrt{D}}</math> with discriminant D. For the conventional Wall–Sun–Sun primes, D = 5. In the general case, a Lucas–Wieferich prime p associated with (P, Q) is a Wieferich prime to base Q and a Wall–Sun–Sun prime with discriminant D = P² − 4Q.<ref name=Elsenhans2010/> In this definition, the prime p should be odd and not divide D.

It is conjectured that for every natural number D, there are infinitely many Wall–Sun–Sun primes with discriminant D.

The case of <math>(P,Q) = (k,-1)</math> corresponds to the k-Wall–Sun–Sun primes, for which Wall–Sun–Sun primes represent the special case k = 1. The k-Wall–Sun–Sun primes can be explicitly defined as primes p such that p² divides the k-Fibonacci number <math>F_k(\pi_k(p))</math>, where F_k(n) = U_n(k, −1) is a Lucas sequence of the first kind with discriminant D = k² + 4 and <math>\pi_k(p)</math> is the Pisano period of k-Fibonacci numbers modulo p.<ref>Template:Cite journal</ref> For a prime p ≠ 2 and not dividing D, this condition is equivalent to either of the following.

• p² divides <math>F_k\left(p - \left(\tfrac{D}{p}\right)\right)</math>, where <math>\left(\tfrac{D}{p}\right)</math> is the Kronecker symbol;
• V_p(k, −1) ≡ k (mod p²), where V_n(k, −1) is a Lucas sequence of the second kind.

The smallest k-Wall–Sun–Sun primes for k = 2, 3, ... are

13, 241, 2, 3, 191, 5, 2, 3, 2683, ... (sequence A271782 in the OEIS)

See also

• Wieferich prime
• Wolstenholme prime
• Wilson prime
• PrimeGrid
• Fibonacci prime
• Pisano period
• Table of congruences

Further reading

• Template:Cite book
• Template:Cite arXiv

External links

• Chris Caldwell, The Prime Glossary: Wall–Sun–Sun prime at the Prime Pages.
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Wall-Sun-SunPrime%7CWall-Sun-SunPrime.html}} |title = Wall–Sun–Sun prime |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

• Richard McIntosh, Status of the search for Wall–Sun–Sun primes (October 2003)
• Template:OEIS el