Document Zbl 0854.11002

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Comput.</a> <a href="/?q=in%3A38355" title="Articles in this Issue">66, No. 217, 433-449 (1997)</a>. </div> <div class="abstract">Summary: An odd prime $p$ is called a Wieferich prime if \[ 2^{p-1} \equiv 1 \pmod{p^{2}}; \] alternatively, a Wilson prime if \[ (p-1)! \equiv -1 \pmod{ p^{2}}. \] To date, the only known Wieferich primes are $p = 1093$ and $3511$, while the only known Wilson primes are $p = 5, 13$, and $563$. We report that there exist no new Wieferich primes $p < 4 \times 10^{12}$, and no new Wilson primes $p < 5 \times 10^{8}$. It is elementary that both defining congruences above hold merely (mod $p$), and it is sometimes estimated on heuristic grounds that the “probability” that $p$ is Wieferich (independently: that $p$ is Wilson) is about $1/p$. We provide some statistical data relevant to occurrences of small values of the pertinent Fermat and Wilson quotients (mod $p$).</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=ci%3A963307">Cited in <strong>5</strong> Reviews</a></div><div class="clear"><a href="/?q=rf%3A963307">Cited in <strong>57</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A11A07" title="MSC2020">11A07</a> </td> <td class="space"> Congruences; primitive roots; residue systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11-04" title="MSC2020">11-04</a> </td> <td class="space"> Software, source code, etc. for problems pertaining to number theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A11Y99" title="MSC2020">11Y99</a> </td> <td class="space"> Computational number theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3AWieferich+primes">Wieferich primes</a>; <a href="/?q=ut%3AWilson+primes">Wilson primes</a>; <a href="/?q=ut%3AFermat+quotients">Fermat quotients</a>; <a href="/?q=ut%3AWilson+quotients">Wilson quotients</a>; <a href="/?q=ut%3Afactorial+evaluation">factorial evaluation</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 0854.11002" data-ciurl="/ci/00963307" data-biburl="/bibtex/00963307.bib" data-amsurl="/amsrefs/00963307.bib" data-xmlurl="/xml/00963307.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/00963307.pdf" title="Zbl 0854.11002 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1090/S0025-5718-97-00791-6" aria-label="DOI for “A search for Wieferich and Wilson primes”" title="10.1090/S0025-5718-97-00791-6">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="mathoverflow"> <h3>Online Encyclopedia of Integer Sequences:</h3> <a href="https://oeis.org/A60371">a(n) = (prime(n) - 1)! + 1.</a><br> <a href="https://oeis.org/A195988">Near-Wieferich primes above 10^9: primes p > 10^9 such that 2^((p-1)/2) == +-1 + A*p (mod p^2) with |A| <= 100, i.e., p=prime(i) such that A258367(i) <= 100.</a><br> <a href="https://oeis.org/A246568">Near-Wieferich primes (primes p satisfying 2^((p-1)/2) == +-1 + A*p (mod p^2)) with |A| < 10.</a><br> <a href="https://oeis.org/A250406">Values of B such that p = prime(n) satisfies (p-1)! == -1-B*p (mod p^2), i.e., p is a near-Wilson prime.</a><br> <a href="https://oeis.org/A250407">Near-Wilson primes (p = prime(n) satisfying (p-1)! == -1-A250406(n)*p (mod p^2)) with A250406(n) < 10.</a><br> <a href="https://oeis.org/A306885">Minimal near-Wieferich A-value (absolute) for all primes in the interval [10^n, 10^(n+1)].</a><br> <a href="https://oeis.org/A353141">Near-Wieferich primes with abs(A) < 2.</a><br> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">T. Agoh, K. Dilcher and L. Skula, Fermat and Wilson quotients for composite moduli, Preprint (1995). · <a href="/1024.11003" class="nowrap">Zbl 1024.11003</a></td> </tr><tr> <td>[2]</td> <td class="space">N. G. W. H. Beeger, Quelques remarques sur les congruences $r^{p-1}\equiv 1\pmod {p^{2}}$ et $(p-1)!\equiv -1\pmod {p^{2}}$, The Messenger of Mathematics 43 (1913-1914), 72-84. · <a href="/44.0227.01" class="nowrap">JFM 44.0227.01</a></td> </tr><tr> <td>[3]</td> <td class="space">B. Berndt, R. Evans and K. Williams, Gauss and Jacobi sums, Wiley-Interscience, to appear. · <a href="/0906.11001" class="nowrap">Zbl 0906.11001</a></td> </tr><tr> <td>[4]</td> <td class="space">Jonathan M. Borwein and Peter B. Borwein, Pi and the AGM, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1987. 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