RECORDS IN COMBINATORICS, NUMBER THEORY AND ALGEBRA

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      • The number of non isomorphic posets with N elements

        P(0) = 1
        P(1) = 1
        P(2) = 2
        P(3) = 5
        P(4) = 16
        P(5) = 63
        P(6) = 318
        P(7) = 2.045 (J.Wright 1972)
        P(8) = 16.999 (S. K. Das 1977)
        P(9) = 183.231 (R. H. Mohring 1984)
        P(10) = 2.567.284 (J. C. Culberson, G. J. E. Rawlins 1990)
        P(11) = 46.749.427 (J. C. Culberson, G. J. E. Rawlins 1990)
        P(12) = 1.104.891.746 (C. Chaunier, N. Lygeros 1991)
        P(13) = 33.823.827.452 (C. Chaunier, N. Lygeros 1992)
        P(14) = 1.338.193.159.771 (N. Lygeros, P. Zimmermann 2000)


      • N consecutive primes in arithmetic progression

        N = 5 (W.J. Blundon, M.F. Jones, M. Lal 1967)
        N = 6 (L. J. Lander, T. R. Parkin 1967)
        N = 7 (H. Dubner, H. Nelson 08/1995)
        N = 8 (H. Dubner, T. Forbes, N. Lygeros, M. Mizony, P. Zimmermann 11/1997)
        N = 9 (H. Dubner, T. Forbes, N. Lygeros, M. Mizony, P. Zimmermann 01/1998)
        Helpers : Quercia (26), Nicholson, Gorham et Mulholland (5.2), Heylen (4.5), Banderier (3.5), Sunde (3.5), Ballinger (3.4), Landreau (3.4), R. Dubner (3.2), Metzner (3.2), Beard (3), Wehmeier (3), Toplic (3), Kierkegaard (2.9), Stevenson (2.5), Biavati (2), Briggs (2), Canart (2), Bronson (1.5), Fauque (1.5), Ketner (1.5), Weiner (1.5), B. et T. Butka (1.4), Taeschner (1.2), Sandbakken (1.2), Rossi (1.2), Buddenhagen (1), O’Hare (1), Peterson (1), Nebe (1), Ruescher (1), Tholberg (1) and Alm, Aschenbrenner, Backhouse, Bell, Bernhard, Berry, Bonacina, Bonn, Clark, Davis, Cohen, Cook, Delepine, Dijkhof, Edge, Erra, Geovanis, Grabysz, Groleau, Gulbrandsen, Hall, Hogan, Jaxon, Kluk, Kriesel, Lamiral, Leunissen, Leong, Leong, Muller, Nastos, Neely, Nunes, Olsen, Polster, Schell, Schroeder, Smart, Smith, Spagnesi, Speare, Svallmark, Trevisan, Weinfurtner, Welsh, Wendling, Womack, Yeackley, Zapata.

 N =10 (H. Dubner, T. Forbes, N. Lygeros, M. Mizony, H. Nelson, P. Zimmermann 03/1998)
Helpers : Piche (6.0), Heylen (4.76), Toplic (4.25), Kierkegaard (3.0), Erra (2.38), J. Lygeros (1.5), Delepine (1.5), Ketner (1.01), Sandbakken (1.0),Taeschner (0.73), Leunissen (0.70), Fauque (0.68), Zapata (0.66), Ballinger (0.58), Helling (0.51) Backhouse (0.50) and Adrian, Alm, Angrist, Aschenbrenner, Banderier, Belshoff, Berkhout, Bernhard, Bosser, Bray, Brown, Carlin, Colligan, Cors, La Curan, Dwyer, Graham, Groleau, Hali, Hogan, Hyvnen, Johannesdal, Johnson, Kibel, Kidd, Kluk, Kotsireas, Kultala, Kriesel, Kveps, Lamiral, Langlois, Megyesi, Olathe, Olly, Palka, Pedersen, Pinch, Pnisch, Pretti, Rasinen, Richard Richter, Sadoway, Serre, Spagnesi, Szeptycki, Tanative4, Thomas, Turner, Valdes, Joe W. W.,Welsh, Weyhaupt.

                  Numbers and Computers (12): Primes in Arithmetic Progression by Albert N. Debono


      • Large Factors Found By Elliptic Curve Method

        Champion of 40 digits
        1232079689567662686148201863995544247703 p(11279) (Lenstra-Dixon 10/1991)
        Champion of 42 digits
        184976479633092931103313037835504355363361 10,201- (D. Rusin 04/1992)
        Champion of 43 digits
        5688864305048653702791752405107044435136231 p(19997) (Berger-Mueller 03/1993)
        Champion of 44 digits
        27885873044042449777540626664487051863162949 p(19069) (Berger-Mueller 06/1995)
        Champion of 47 digits
        12025702000065183805751513732616276516181800961 5,256+ (P. Montgomery 11/1995)
        28207978317787299519881883345010831781124600233 30,109- (P. Montgomery 02/1996)
        Champion of 48 digits
        662926550178509475639682769961460088456141816377 24,121+ (R. P. Brent 10/1997)
        Champion of 49 digits
        1078825191548640568143407841173742460493739682993 2,1071+ (P. Zimmermann 06/1998)
        Champion of 53 digits
        53625112691923843508117942311516428173021903300344567 2,677- (C. Curry 09/1998)
        Champion of 54 digits
        484061254276878368125726870789180231995964870094916937 (N. Lygeros, M. Mizony 12/1999)
        For more information see ecmnet.


      • First Erdös-Woods Numbers.For d<=520 see Cégielski and al.
        16,22,34,36,46,56,64,66,70,76,78,86,88,92,94,96,
        100,106,112,116,118,120,124,130,134,142,144,146,154,160,162,186,190,196,
        204,210,216,218,220,222,232,238,246,248,250,256,260,262,268,276,280,286,288,292,296,298,
        300,302,306,310,316,320,324,326,328,330,336,340,342,346,356,366,372,378,382,394,396,
        400,404,406,408,414,416,424,426,428,430,438,446,454,456,466,470,472,474,476,484,486,490,494,498,
        512,516,518,520,526,528,532,534,536,538,540,546,550,552,554,556,560,574,576,580,582,584,590,
        604,606,612,616,624,630,634,636,640,650,666,668,670,
      • Consecutive Erdös-Woods Numbers.
        N=2: 34,36
        N=3: 92,94,96
        N=4: 216,218,220,222 (P. Cégielski, D.Richard 2000)
        N=5: 532,534,536,538,540 (P. Cégielski, D.Richard 2000)
        N=6: 1834,1836,1838,1840,1842,1844 (N. Lygeros 2001)
        N=7: 2166,2168,2170,2172,2174,2176,2178 (N. Lygeros 2001)
        N=8: 4312,4314,4316,4318,4320,4322,4324,4326 (N. Lygeros 2001)
        N=9: 4032,4034,4036,4038,4040,4042,4044,4046,4048 (N. Lygeros 2001)
        For more information see Erdös-Woods Project.


      • Circle orders and SSU conjecture.

        N=07 : (N. Lygeros 1991)
        N=08 : (R. Bayon, N. Lygeros 10/2001)
        N=09 : (R. Bayon, N. Lygeros, J.-S. Sereni 02/2002)
        N=10 : (R. Bayon, N. Lygeros, J.-S. Sereni 06/2002)
        HELPERS : Philippe Alsina, Lionel Baraque, Bruno Carré, Patrice Deloche, Pierre Hyvernat, RĂ©mi Nardini, Matthieu Pérotin, Romain Perron.


      • The number of non isomorphic mixed models with factors

        f=1 : 2 (A. Hess, H. Iyer 1999)
        f=2 : 6 (A. Hess, H. Iyer 1999)
        f=3 : 22 (A. Hess, H. Iyer 1999)
        f=4 : 101 (A. Hess, H. Iyer 1999)
        f=5 : 576 (A. Hess, H. Iyer 1999)
        f=6 : 4.162 (R.Bayon, N. Lygeros and J.-S. Sereni 2002)
        f=7 : 38.280 (R.Bayon, N. Lygeros and J.-S. Sereni 2002)
        f=8 : 451.411 (R.Bayon, N. Lygeros and J.-S. Sereni 2002)
        f=9 : 6.847.662 (R.Bayon, N. Lygeros and J.-S. Sereni 2002)
        For more information see Mixed Models Project


      • The number of abelian hypergroups of order N.

        N=02 : 6 (R. Bayon, N. Lygeros 10/2004)
        N=03 : 466 (R. Bayon, N. Lygeros 10/2004)
        N=04 : 10.614.362 (R. Bayon, N. Lygeros 03/2005)
        HELPERS : Philippe Alsina, Patrice Deloche, Yoann Martinez.


      • The number of abelian Hv-groups of order N.
        N=02 : 10 (R. Bayon, N. Lygeros 10/2004)
        N=03 : 7.926 (R. Bayon, N. Lygeros 10/2004)
        N=04 : 8.028.299.905 (R. Bayon, N. Lygeros 04/2005)
        HELPERS : Philippe Alsina, Patrice Deloche, Yoann Martinez.


      • The number of hyperrings of order N.
        N=02 :63 (R. Bayon, N. Lygeros 2007)
        N=03 : 33 277 642 (R. Bayon, N. Lygeros 2007)


      • The number of Hv-rings of order N.
        N=02 : 875 (R. Bayon, N. Lygeros 2007)


      • Solutions to the Equation τ(p) = 0 (mod p)
        N=01 :  2 (S. Ramanujan)
        N=02 : 3 (S. Ramanujan)
        N=03 : 5 (S. Ramanujan)
        N=04 : 7 (S. Ramanujan)
        N=05 : 2411 (Newman)
        N=06 : 7758337633 (N. Lygeros, O. Rozier  03/2010)
        For more information see Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.4


      • Certified Primes Numbers with Elliptic Curves Method
        20562 Digits. F. Morain (06/2005)
        25050 Digits. P. Leyland, F. Morain (10/2010)
        26643 Digits. N. Lygeros, F. Morain, O. Rozier (03/2011)
        29492 Digits. N. Lygeros, O. Rozier (09/2015)
        HELPERS : Philippe Alsina, Pierre Gazzano, André Solaris Odd prime values of the Ramanujan tau function in The Ramanujan Journal , 15/03/2013– Pdf


      • Large Factors Found By Elliptic Curve Method
        Champion of 54 digits.
        113944651856655107794996103150041939333993926230123191 (N. Lygeros, M. Mizony 03/2000)
        For more information see ecmnet