Wieferich prime (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Wieferich prime" in English language version.

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31doi.org

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12ams.org

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7harvard.edu

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6jstor.org

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5uni-goettingen.de

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4zbmath.org

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4books.google.com

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4fermatquotient.com

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ams.org

McIntosh, R.J.; Roettger, E.L. (2007), "A search for Fibonacci–Wieferich and Wolstenholme primes" (PDF), Mathematics of Computation, 76 (260): 2087–2094, Bibcode:2007MaCom..76.2087M, CiteSeerX 10.1.1.105.9393, doi:10.1090/S0025-5718-07-01955-2
Wilfrid Keller; Jörg Richstein (2005), "Solutions of the congruence $a p -1 \equiv 1 (mod p r)$ " (PDF), Mathematics of Computation, 74 (250): 927–936, doi:10.1090/S0025-5718-04-01666-7.
Kravitz, S. (1960). "The Congruence 2^p-1 ≡ 1 (mod p²) for p < 100,000" (PDF). Mathematics of Computation. 14 (72): 378. doi:10.1090/S0025-5718-1960-0121334-7.
Fröberg C. E. (1958). "Some Computations of Wilson and Fermat Remainders" (PDF). Mathematics of Computation. 12 (64): 281. doi:10.1090/S0025-5718-58-99270-6.
Riesel, H. (1964). "Note on the Congruence a^p−1 ≡ 1 (mod p²)" (PDF). Mathematics of Computation. 18 (85): 149–150. doi:10.1090/S0025-5718-1964-0157928-6.
Lehmer, D. H. (1981). "On Fermat's quotient, base two" (PDF). Mathematics of Computation. 36 (153): 289–290. doi:10.1090/S0025-5718-1981-0595064-5.
Coppersmith, D. (1990), "Fermat's Last Theorem (Case I) and the Wieferich Criterion" (PDF), Mathematics of Computation, 54 (190): 895–902, Bibcode:1990MaCom..54..895C, doi:10.1090/s0025-5718-1990-1010598-2, JSTOR 2008518.
Cikánek, P. (1994), "A Special Extension of Wieferich's Criterion" (PDF), Mathematics of Computation, 62 (206): 923–930, Bibcode:1994MaCom..62..923C, doi:10.2307/2153550, JSTOR 3562296.
Dilcher, K.; Skula, L. (1995), "A new criterion for the first case of Fermat's last theorem" (PDF), Mathematics of Computation, 64 (209): 363–392, Bibcode:1995MaCom..64..363D, doi:10.1090/s0025-5718-1995-1248969-6, JSTOR 2153341
Jakubec, S. (1998), "On divisibility of the class number h⁺ of the real cyclotomic fields of prime degree l" (PDF), Mathematics of Computation, 67 (221): 369–398, doi:10.1090/s0025-5718-98-00916-8
Joshua Knauer; Jörg Richstein (2005), "The continuing search for Wieferich primes" (PDF), Mathematics of Computation, 74 (251): 1559–1563, Bibcode:2005MaCom..74.1559K, doi:10.1090/S0025-5718-05-01723-0.

mathscinet.ams.org

Bray, H. G.; Warren, L. J. (1967), "On the square-freeness of Fermat and Mersenne numbers", Pacific J. Math., 22 (3): 563–564, doi:10.2140/pjm.1967.22.563, MR 0220666, Zbl 0149.28204

archive.org

Meyer, W. Fr. (1902). "Ergänzungen zum Fermatschen und Wilsonschen Satze". Arch. Math. Physik. 3. 2: 141–146. Retrieved 2020-09-02.
Beeger, N. G. W. H. (1922), "On a new case of the congruence 2^{p − 1} ≡ 1 (mod p²)", Messenger of Mathematics, 51: 149–150

books.google.com

Ribenboim, Paulo (2004), Die Welt der Primzahlen: Geheimnisse und Rekorde (in German), New York: Springer, p. 237, ISBN 978-3-540-34283-0
Ribenboim, P. (1979). 13 Lectures on Fermat's Last Theorem. New York: Springer. p. 154. ISBN 978-0-387-90432-0.
Ribenboim, Paulo (1991), The little book of big primes, New York: Springer, p. 64, ISBN 978-0-387-97508-5
Müller, H. (2009). "Über Periodenlängen und die Vermutungen von Collatz und Crandall". Mitteilungen der Mathematischen Gesellschaft in Hamburg (in German). 28: 121–130.

dartmouth.edu

math.dartmouth.edu

Franco, Z.; Pomerance, C. (1995), "On a conjecture of Crandall concerning the qx + 1 problem" (PDF), Mathematics of Computation, 64 (211): 1333–36, Bibcode:1995MaCom..64.1333F, doi:10.2307/2153499, JSTOR 2153499.
Leonhard Euler (1736), "Theorematum quorundam ad numeros primos spectantium demonstratio" (PDF), Novi Comm. Acad. Sci. Petropol. (in Latin), 8: 33–37.

gauss.dartmouth.edu

Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes" (PDF), Mathematics of Computation, 66 (217): 433–449, Bibcode:1997MaCom..66..433C, doi:10.1090/S0025-5718-97-00791-6.

digizeitschriften.de

Wells Johnson (1977), "On the nonvanishing of Fermat quotients $(mod p)$ ", J. Reine Angew. Math., 292: 196–200

doi.org

Franco, Z.; Pomerance, C. (1995), "On a conjecture of Crandall concerning the qx + 1 problem" (PDF), Mathematics of Computation, 64 (211): 1333–36, Bibcode:1995MaCom..64.1333F, doi:10.2307/2153499, JSTOR 2153499.
Banks, W.D.; Luca, F.; Shparlinski, I.E. (2007), "Estimates for Wieferich numbers" (PDF), The Ramanujan Journal, 14 (3): 361–378, doi:10.1007/s11139-007-9030-z, S2CID 39279379, archived from the original (PDF) on 2013-05-03, retrieved 2011-03-12.
McIntosh, R.J.; Roettger, E.L. (2007), "A search for Fibonacci–Wieferich and Wolstenholme primes" (PDF), Mathematics of Computation, 76 (260): 2087–2094, Bibcode:2007MaCom..76.2087M, CiteSeerX 10.1.1.105.9393, doi:10.1090/S0025-5718-07-01955-2
Israel Kleiner (2000), "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem", Elemente der Mathematik, 55: 21, doi:10.1007/PL00000079, S2CID 53319514.
Dickson, L. E. (1917), "Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers", Annals of Mathematics, 18 (4): 161–187, doi:10.2307/2007234, JSTOR 2007234
Wilfrid Keller; Jörg Richstein (2005), "Solutions of the congruence $a p -1 \equiv 1 (mod p r)$ " (PDF), Mathematics of Computation, 74 (250): 927–936, doi:10.1090/S0025-5718-04-01666-7.
Wieferich, A. (1909), "Zum letzten Fermat'schen Theorem", Journal für die reine und angewandte Mathematik (in German), 1909 (136): 293–302, doi:10.1515/crll.1909.136.293, S2CID 118715277.
Ribenboim, P. (1983), "1093", The Mathematical Intelligencer, 5 (2): 28–34, doi:10.1007/BF03023623
Guy, R. K. (1965), "A property of the prime 3511", The Mathematical Gazette, 49 (367): 78–79, doi:10.2307/3614249, JSTOR 3614249
Kravitz, S. (1960). "The Congruence 2^p-1 ≡ 1 (mod p²) for p < 100,000" (PDF). Mathematics of Computation. 14 (72): 378. doi:10.1090/S0025-5718-1960-0121334-7.
Fröberg C. E. (1958). "Some Computations of Wilson and Fermat Remainders" (PDF). Mathematics of Computation. 12 (64): 281. doi:10.1090/S0025-5718-58-99270-6.
Riesel, H. (1964). "Note on the Congruence a^p−1 ≡ 1 (mod p²)" (PDF). Mathematics of Computation. 18 (85): 149–150. doi:10.1090/S0025-5718-1964-0157928-6.
Lehmer, D. H. (1981). "On Fermat's quotient, base two" (PDF). Mathematics of Computation. 36 (153): 289–290. doi:10.1090/S0025-5718-1981-0595064-5.
Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes" (PDF), Mathematics of Computation, 66 (217): 433–449, Bibcode:1997MaCom..66..433C, doi:10.1090/S0025-5718-97-00791-6.
Coppersmith, D. (1990), "Fermat's Last Theorem (Case I) and the Wieferich Criterion" (PDF), Mathematics of Computation, 54 (190): 895–902, Bibcode:1990MaCom..54..895C, doi:10.1090/s0025-5718-1990-1010598-2, JSTOR 2008518.
Cikánek, P. (1994), "A Special Extension of Wieferich's Criterion" (PDF), Mathematics of Computation, 62 (206): 923–930, Bibcode:1994MaCom..62..923C, doi:10.2307/2153550, JSTOR 3562296.
Dilcher, K.; Skula, L. (1995), "A new criterion for the first case of Fermat's last theorem" (PDF), Mathematics of Computation, 64 (209): 363–392, Bibcode:1995MaCom..64..363D, doi:10.1090/s0025-5718-1995-1248969-6, JSTOR 2153341
Granville, A.; Monagan, M. B. (1988), "The First Case of Fermat's Last Theorem is true for all prime exponents up to 714,591,416,091,389", Transactions of the American Mathematical Society, 306 (1): 329–359, doi:10.1090/S0002-9947-1988-0927694-5.
Suzuki, Jiro (1994), "On the generalized Wieferich criteria", Proceedings of the Japan Academy, Series A, 70 (7): 230–234, doi:10.3792/pjaa.70.230
Silverman, J. H. (1988), "Wieferich's criterion and the abc-conjecture", Journal of Number Theory, 30 (2): 226–237, doi:10.1016/0022-314X(88)90019-4
Bray, H. G.; Warren, L. J. (1967), "On the square-freeness of Fermat and Mersenne numbers", Pacific J. Math., 22 (3): 563–564, doi:10.2140/pjm.1967.22.563, MR 0220666, Zbl 0149.28204
Scott, R.; Styer, R. (April 2004). "On $p x - q y = c$ and related three term exponential Diophantine equations with prime bases". Journal of Number Theory. 105 (2): 212–234. doi:10.1016/j.jnt.2003.11.008.
Scott, R.; Styer, R. (2006). "On the generalized Pillai equation ±a^x±b^y = c". Journal of Number Theory. 118 (2): 236–265. doi:10.1016/j.jnt.2005.09.001.
Dobeš, Jan; Kureš, Miroslav (2010). "Search for Wieferich primes through the use of periodic binary strings". Serdica Journal of Computing. 4 (3): 293–300. doi:10.55630/sjc.2010.4.293-300. hdl:10525/1595. Zbl 1246.11019.
Pinch, R. G. E. (2000). The Pseudoprimes up to 10¹³. Lecture Notes in Computer Science. Vol. 1838. pp. 459–473. doi:10.1007/10722028_30. ISBN 978-3-540-67695-9.
Ehrlich, A. (1994), "Cycles in Doubling Diagrams mod m" (PDF), The Fibonacci Quarterly, 32 (1): 74–78, doi:10.1080/00150517.1994.12429259.
Jakubec, S. (1995), "Connection between the Wieferich congruence and divisibility of h⁺" (PDF), Acta Arithmetica, 71 (1): 55–64, doi:10.4064/aa-71-1-55-64
Jakubec, S. (1998), "On divisibility of the class number h⁺ of the real cyclotomic fields of prime degree l" (PDF), Mathematics of Computation, 67 (221): 369–398, doi:10.1090/s0025-5718-98-00916-8
Joshua Knauer; Jörg Richstein (2005), "The continuing search for Wieferich primes" (PDF), Mathematics of Computation, 74 (251): 1559–1563, Bibcode:2005MaCom..74.1559K, doi:10.1090/S0025-5718-05-01723-0.
Agoh, T.; Dilcher, K.; Skula, L. (1997), "Fermat Quotients for Composite Moduli", Journal of Number Theory, 66 (1): 29–50, doi:10.1006/jnth.1997.2162
Voloch, J. F. (2000), "Elliptic Wieferich Primes", Journal of Number Theory, 81 (2): 205–209, doi:10.1006/jnth.1999.2471

elmath.org

"statistics". elMath.org. 2016-09-02. Archived from the original on 2016-09-02. Retrieved 2019-09-18.
"About project Wieferich@Home". Archived from the original on 2012-03-22. Retrieved 2010-07-05.

elte.hu

ac.inf.elte.hu

DeKoninck, J.-M.; Doyon, N. (2007), "On the set of Wieferich primes and of its complement" (PDF), Annales Univ. Sci. Budapest., Sect. Comp., 27: 3–13

fermatquotient.com

"Wieferich primes to base 1052". Archived from the original on 2015-09-11. Retrieved 2014-07-22.
"Wieferich primes to base 10125".
"Wieferich primes with level ≥ 3".
"Solution of $(a + 1) p -1 - a p -1 \equiv 0 (mod p 2)$ ".

files.wordpress.com

trex58.files.wordpress.com

Ribenboim, P. (2004). "Chapter 2. How to Recognize Whether a Natural Number is a Prime" (PDF). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 99. ISBN 978-0-387-20169-6.

handle.net

hdl.handle.net

Dobeš, Jan; Kureš, Miroslav (2010). "Search for Wieferich primes through the use of periodic binary strings". Serdica Journal of Computing. 4 (3): 293–300. doi:10.55630/sjc.2010.4.293-300. hdl:10525/1595. Zbl 1246.11019.

harvard.edu

ui.adsabs.harvard.edu

Franco, Z.; Pomerance, C. (1995), "On a conjecture of Crandall concerning the qx + 1 problem" (PDF), Mathematics of Computation, 64 (211): 1333–36, Bibcode:1995MaCom..64.1333F, doi:10.2307/2153499, JSTOR 2153499.
McIntosh, R.J.; Roettger, E.L. (2007), "A search for Fibonacci–Wieferich and Wolstenholme primes" (PDF), Mathematics of Computation, 76 (260): 2087–2094, Bibcode:2007MaCom..76.2087M, CiteSeerX 10.1.1.105.9393, doi:10.1090/S0025-5718-07-01955-2
Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997), "A search for Wieferich and Wilson primes" (PDF), Mathematics of Computation, 66 (217): 433–449, Bibcode:1997MaCom..66..433C, doi:10.1090/S0025-5718-97-00791-6.
Coppersmith, D. (1990), "Fermat's Last Theorem (Case I) and the Wieferich Criterion" (PDF), Mathematics of Computation, 54 (190): 895–902, Bibcode:1990MaCom..54..895C, doi:10.1090/s0025-5718-1990-1010598-2, JSTOR 2008518.
Cikánek, P. (1994), "A Special Extension of Wieferich's Criterion" (PDF), Mathematics of Computation, 62 (206): 923–930, Bibcode:1994MaCom..62..923C, doi:10.2307/2153550, JSTOR 3562296.
Dilcher, K.; Skula, L. (1995), "A new criterion for the first case of Fermat's last theorem" (PDF), Mathematics of Computation, 64 (209): 363–392, Bibcode:1995MaCom..64..363D, doi:10.1090/s0025-5718-1995-1248969-6, JSTOR 2153341
Joshua Knauer; Jörg Richstein (2005), "The continuing search for Wieferich primes" (PDF), Mathematics of Computation, 74 (251): 1559–1563, Bibcode:2005MaCom..74.1559K, doi:10.1090/S0025-5718-05-01723-0.

icm.edu.pl

matwbn.icm.edu.pl

Jakubec, S. (1995), "Connection between the Wieferich congruence and divisibility of h⁺" (PDF), Acta Arithmetica, 71 (1): 55–64, doi:10.4064/aa-71-1-55-64

jstor.org

Franco, Z.; Pomerance, C. (1995), "On a conjecture of Crandall concerning the qx + 1 problem" (PDF), Mathematics of Computation, 64 (211): 1333–36, Bibcode:1995MaCom..64.1333F, doi:10.2307/2153499, JSTOR 2153499.
Dickson, L. E. (1917), "Fermat's Last Theorem and the Origin and Nature of the Theory of Algebraic Numbers", Annals of Mathematics, 18 (4): 161–187, doi:10.2307/2007234, JSTOR 2007234
Guy, R. K. (1965), "A property of the prime 3511", The Mathematical Gazette, 49 (367): 78–79, doi:10.2307/3614249, JSTOR 3614249
Coppersmith, D. (1990), "Fermat's Last Theorem (Case I) and the Wieferich Criterion" (PDF), Mathematics of Computation, 54 (190): 895–902, Bibcode:1990MaCom..54..895C, doi:10.1090/s0025-5718-1990-1010598-2, JSTOR 2008518.
Cikánek, P. (1994), "A Special Extension of Wieferich's Criterion" (PDF), Mathematics of Computation, 62 (206): 923–930, Bibcode:1994MaCom..62..923C, doi:10.2307/2153550, JSTOR 3562296.
Dilcher, K.; Skula, L. (1995), "A new criterion for the first case of Fermat's last theorem" (PDF), Mathematics of Computation, 64 (209): 363–392, Bibcode:1995MaCom..64..363D, doi:10.1090/s0025-5718-1995-1248969-6, JSTOR 2153341

kaist.ac.kr

basilo.kaist.ac.kr

Byeon, D. (2006), "Class numbers, Iwasawa invariants and modular forms" (PDF), Trends in Mathematics, 9 (1): 25–29, archived from the original (PDF) on 2012-04-26, retrieved 2012-09-05

math.ca

fq.math.ca

Ehrlich, A. (1994), "Cycles in Doubling Diagrams mod m" (PDF), The Fibonacci Quarterly, 32 (1): 74–78, doi:10.1080/00150517.1994.12429259.

mq.edu.au

web.science.mq.edu.au

Banks, W.D.; Luca, F.; Shparlinski, I.E. (2007), "Estimates for Wieferich numbers" (PDF), The Ramanujan Journal, 14 (3): 361–378, doi:10.1007/s11139-007-9030-z, S2CID 39279379, archived from the original (PDF) on 2013-05-03, retrieved 2011-03-12.

oeis.org

Meissner, W. (1913), "Über die Teilbarkeit von 2^p − 2 durch das Quadrat der Primzahl p=1093" (PDF), Sitzungsber. D. Königl. Preuss. Akad. D. Wiss. (in German), Zweiter Halbband. Juli bis Dezember, Berlin: 663–667, JFM 44.0218.02

primegrid.com

"WSS and WFS are suspended". PrimeGrid Message Board. May 11, 2017.
"Message boards : Wieferich and Wall-Sun-Sun Prime Search". PrimeGrid.
"WW Statistics". PrimeGrid.
PrimeGrid, Wieferich & near Wieferich primes p < 11e15 Archived 2012-10-18 at the Wayback Machine

projecteuclid.org

Suzuki, Jiro (1994), "On the generalized Wieferich criteria", Proceedings of the Japan Academy, Series A, 70 (7): 230–234, doi:10.3792/pjaa.70.230
Bray, H. G.; Warren, L. J. (1967), "On the square-freeness of Fermat and Mersenne numbers", Pacific J. Math., 22 (3): 563–564, doi:10.2140/pjm.1967.22.563, MR 0220666, Zbl 0149.28204

psu.edu

citeseerx.ist.psu.edu

McIntosh, R.J.; Roettger, E.L. (2007), "A search for Fibonacci–Wieferich and Wolstenholme primes" (PDF), Mathematics of Computation, 76 (260): 2087–2094, Bibcode:2007MaCom..76.2087M, CiteSeerX 10.1.1.105.9393, doi:10.1090/S0025-5718-07-01955-2

pte.hu

ttk.pte.hu

Kiss, E.; Sándor, J. (2004). "On a congruence by János Bolyai, connected with pseudoprimes" (PDF). Mathematica Pannonica. 15 (2): 283–288.

researchgate.net

Dobeš, Jan; Kureš, Miroslav (2010). "Search for Wieferich primes through the use of periodic binary strings". Serdica Journal of Computing. 4 (3): 293–300. doi:10.55630/sjc.2010.4.293-300. hdl:10525/1595. Zbl 1246.11019.

semanticscholar.org

api.semanticscholar.org

Banks, W.D.; Luca, F.; Shparlinski, I.E. (2007), "Estimates for Wieferich numbers" (PDF), The Ramanujan Journal, 14 (3): 361–378, doi:10.1007/s11139-007-9030-z, S2CID 39279379, archived from the original (PDF) on 2013-05-03, retrieved 2011-03-12.
Israel Kleiner (2000), "From Fermat to Wiles: Fermat's Last Theorem Becomes a Theorem", Elemente der Mathematik, 55: 21, doi:10.1007/PL00000079, S2CID 53319514.
Wieferich, A. (1909), "Zum letzten Fermat'schen Theorem", Journal für die reine und angewandte Mathematik (in German), 1909 (136): 293–302, doi:10.1515/crll.1909.136.293, S2CID 118715277.

uni-goettingen.de

gdz.sub.uni-goettingen.de

Wieferich, A. (1909), "Zum letzten Fermat'schen Theorem", Journal für die reine und angewandte Mathematik (in German), 1909 (136): 293–302, doi:10.1515/crll.1909.136.293, S2CID 118715277.
Haentzschel, E. (1916), "Über die Kongruenz 2¹⁰⁹² ≡ 1 (mod 1093²)", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 25: 284
Haentzschel, E. (1925), "Über die Kongruenz 2¹⁰⁹² ≡ 1 (mod 1093²)", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 34: 184

resolver.sub.uni-goettingen.de

Bachmann, P. (1913). "Über den Rest von ${\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p$ ". Journal für Mathematik (in German). 142 (1): 41–50.
Rotkiewicz, A. (1965). "Sur les nombres de Mersenne dépourvus de diviseurs carrés et sur les nombres naturels n, tels que $n 2 |2 n - 2$ ". Mat. Vesnik (in French). 2 (17): 78–80.

uni-hamburg.de

www1.uni-hamburg.de

"Fermat quotients q_p(a) that are divisible by p". www1.uni-hamburg.de. 2014-08-09. Archived from the original on 2014-08-09. Retrieved 2019-09-18.

utm.edu

primes.utm.edu

The Prime Glossary: Wieferich prime
Mersenne Primes: Conjectures and Unsolved Problems
Fermat Quotient at The Prime Glossary

uwaterloo.ca

cs.uwaterloo.ca

Dorais, F. G.; Klyve, D. (2011). "A Wieferich Prime Search Up to 6.7×10¹⁵" (PDF). Journal of Integer Sequences. 14 (9). Zbl 1278.11003. Retrieved 2011-10-23.

waikato.ac.nz

math.waikato.ac.nz

Broughan, K. (2006), "Relaxations of the ABC Conjecture using integer k'th roots" (PDF), New Zealand J. Math., 35 (2): 121–136

web.archive.org

Banks, W.D.; Luca, F.; Shparlinski, I.E. (2007), "Estimates for Wieferich numbers" (PDF), The Ramanujan Journal, 14 (3): 361–378, doi:10.1007/s11139-007-9030-z, S2CID 39279379, archived from the original (PDF) on 2013-05-03, retrieved 2011-03-12.
"statistics". elMath.org. 2016-09-02. Archived from the original on 2016-09-02. Retrieved 2019-09-18.
Byeon, D. (2006), "Class numbers, Iwasawa invariants and modular forms" (PDF), Trends in Mathematics, 9 (1): 25–29, archived from the original (PDF) on 2012-04-26, retrieved 2012-09-05
"About project Wieferich@Home". Archived from the original on 2012-03-22. Retrieved 2010-07-05.
PrimeGrid, Wieferich & near Wieferich primes p < 11e15 Archived 2012-10-18 at the Wayback Machine
"Wieferich primes to base 1052". Archived from the original on 2015-09-11. Retrieved 2014-07-22.
"Fermat quotients q_p(a) that are divisible by p". www1.uni-hamburg.de. 2014-08-09. Archived from the original on 2014-08-09. Retrieved 2019-09-18.

wisc.edu

pages.cs.wisc.edu

Charles, D. X. "On Wieferich primes" (PDF). wisc.edu.

wolfram.com

mathworld.wolfram.com

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