Equal Sums of Like Powers

Records on the Generalization of Prouhet-Tarry-Escott Problem

• Ideal non-negative integer solutions ( T/L = 164 types )

Type	First solution found in	First solution found by
( k = 1 )
( k = 2 )
( k = 3 )
( k = 4 )	1772	Euler
( k = 1, 2 )
( k = 1, 3 )	1939	A.Moessner
( k = 1, 4 )	1995-02-08	Chen Shuwen
( k = 1, 5 )	1939	A.Moessner
( k = 2, 3 )	1949	A.Golden
( k = 2, 4 )
( k = 2, 6 )	1934	Subba-Rao
( k = 1, 2, 3 )	1910	L.E.Dickson
( k = 1, 2, 4 )	1995-02-24	Chen Shuwen
( k = 1, 2, 5 )	1995-10-20	Chen Shuwen
( k = 1, 2, 6 )	1995-04-22	L.Lander, T.Parkin, J.Selfridge, Chen Shuwen
( k = 1, 3, 4 )	1995-10-22	Chen Shuwen
( k = 1, 3, 5 )
( k = 1, 3, 7 )	1999	Ajai Choudhry
( k = 2, 3, 4 )	1995-11-15	Chen Shuwen
( k = 2, 4, 6 )	1913	Crussol
( k = 1, 2, 3, 4 )
( k = 1, 2, 3, 5 )	1995-04-09	Chen Shuwen
( k = 1, 2, 3, 6 )	1999-07-04	Chen Shuwen
( k = 1, 2, 3, 7 )	2022-09-15	Chen Shuwen
( k = 1, 2, 4, 6 )	1953	G.Palama
( k = 1, 3, 5, 7 )	1940	A.Golden
( k = 2, 4, 6, 8 )	1940	A.Letac
( k = 1, 2, 3, 4, 5 )	1912	G.Tarry
( k = 1, 2, 3, 4, 6 )	1995-10-27	Chen Shuwen
( k = 1, 2, 3, 4, 7 )	2022-09-22	Chen Shuwen
( k = 1, 2, 3, 5, 7 )	1999-08-16	Chen Shuwen
( k = 1, 2, 4, 6, 8 )	1995-02-27	Chen Shuwen
( k = 1, 3, 5, 7, 9 )	2000-10-16	Chen Shuwen
( k = 2, 4, 6, 8, 10 )	1999-09-07	Nuutti Kuosa, Jean-Charles Meyrignac, Chen Shuwen
( k = 1, 2, 3, 4, 5, 6 )	1910	E.B.Escott
( k = 1, 2, 3, 4, 5, 7 )	1995-11-30	Chen Shuwen
( k = 1, 2, 3, 5, 7, 9 )	2023-01-27	Chen Shuwen
( k = 1, 2, 3, 4, 5, 6, 7 )	1913	G.Tarry
( k = 1, 2, 3, 4, 5, 6, 8 )	1999-06-28	Chen Shuwen
( k = 1, 2, 3, 4, 5, 6, 7, 8 )	1940	A.Letac
( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 )	1940	A.Letac
( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 )	1999-09-07	Nuutti Kuosa, Jean-Charles Meyrignac, Chen Shuwen
( k = -1 )
( k = -2 )	2001-03-26	Chen Shuwen
( k = -3 )	2001-03-26	Chen Shuwen
( k = -4 )	2001-03-26	Euler, Chen Shuwen
( k = -2, -1 )	2001-03-26	Chen Shuwen
( k = -3, -1 )	2001-03-26	A.Moessner, Chen Shuwen
( k = -4, -1 )	2001-03-26	Chen Shuwen
( k = -5, -1 )	2001-03-26	A.Moessner, Chen Shuwen
( k = -3, -2 )	2001-03-26	A.Golden, Chen Shuwen
( k = -4, -2 )	2001-03-26	Chen Shuwen
( k = -6, -2 )	2001-03-26	Subba-Rao, Chen Shuwen
( k = -3, -2, -1 )	2001-03-26	L.E.Dickson, Chen Shuwen
( k = -4, -2, -1 )	2001-03-26	Chen Shuwen
( k = -5, -2, -1 )	2001-03-26	Chen Shuwen
( k = -6, -2, -1 )	2001-03-26	L.Lander, T.Parkin, J.Selfridge, Chen Shuwen
( k = -4, -3, -1 )	2001-03-26	Chen Shuwen
( k = -5, -3, -1 )	2001-03-26	Chen Shuwen
( k = -7, -3, -1 )	2001-03-26	Ajai Choudhry, Chen Shuwen
( k = -4, -3, -2 )	2001-03-26	Chen Shuwen
( k = -6, -4, -2 )	2001-03-26	Crussol, Chen Shuwen
( k = -4, -3, -2, -1 )	2001-03-26	Chen Shuwen
( k = -5, -3, -2, -1 )	2001-03-26	Chen Shuwen
( k = -6, -3, -2, -1 )	2001-03-26	Chen Shuwen
( k = -7, -3, -2, -1 )	2022-09-15	Chen Shuwen
( k = -6, -4, -2, -1 )	2001-03-26	G.Palama, Chen Shuwen
( k = -7, -5, -3, -1 )	2001-03-26	A.Golden, Chen Shuwen
( k = -8, -6, -4, -2 )	2001-03-26	A.Letac, Chen Shuwen
( k = -5, -4, -3, -2, -1 )	2001-03-26	G.Tarry, Chen Shuwen
( k = -7, -5, -3, -2, -1 )	2001-03-26	Chen Shuwen
( k = -6, -4, -3, -2, -1 )	2001-03-26	Chen Shuwen
( k = -7, -4, -3, -2, -1 )	2022-09-22	Chen Shuwen
( k = -8, -6, -4, -2, -1 )	2001-03-26	Chen Shuwen
( k = -9, -7, -5, -3, -1 )	2001-03-26	Chen Shuwen
( k = -10, -8, -6, -4, -2 )	2001-03-26	Nuutti Kuosa, Jean-Charles Meyrignac, Chen Shuwen
( k = -6, -5, -4, -3, -2, -1 )	2001-03-26	E.B.Escott, Chen Shuwen
( k = -7, -5, -4, -3, -2, -1 )	2001-03-26	Chen Shuwen
( k = -9, -7, -5, -3, -2, -1 )	2023-01-27	Chen Shuwen
( k = -7, -6, -5, -4, -3, -2, -1 )	2001-03-26	G.Tarry, Chen Shuwen
( k = -8, -6, -5, -4, -3, -2, -1 )	2001-03-26	Chen Shuwen
( k = -8, -7, -6, -5, -4, -3, -2, -1 )	2001-03-26	A.Letac, Chen Shuwen
( k = -9, -8, -7, -6, -5, -4, -3, -2, -1 )	2001-03-26	A.Letac, Chen Shuwen
( k = -11, -9, -8, -7, -6, -5, -4, -3, -2, -1 )	2001-03-26	Nuutti Kuosa, Jean-Charles Meyrignac, Chen Shuwen
( k = 0 )
( k = 0, 1 )	1939	A.Moessner
( k = 0, 2 )		Bini, Dubouis and Mathieu [49]
( k = 0, 3 )		Stephane Vandemergel [49]
( k = 0, 4 )		Stephane Vandemergel [49]
( k = 0, 1, 2 )	1944	A.Golden
( k = 0, 1, 3 )	2001-05-08	Chen Shuwen
( k = 0, 1, 4 )	2017-03-24	Chen Shuwen
( k = 0, 1, 5 )	2017-03-24	Chen Shuwen
( k = 0, 2, 3 )	2022-10-13	Chen Shuwen
( k = 0, 2, 4 )	2001-05-08	Chen Shuwen
( k = 0, 2, 6 )	2017-05-19	Chen Shuwen
( k = 0, 1, 2, 3 )	2000	Guo Xianqiang, Chen Shuwen
( k = 0, 1, 2, 4 )	2017-03-22	Chen Shuwen
( k = 0, 1, 2, 6 )	2017-05-31	Chen Shuwen
( k = 0, 2, 4, 6 )	2017-03-21	Chen Shuwen
( k = 0, 1, 2, 3, 4 )	2001-03-29	Chen Shuwen
( k = 0, 1, 2, 3, 5 )	2017-04-15	Chen Shuwen
( k = 0, 1, 2, 4, 6 )	2017-03-23	Chen Shuwen
( k = 0, 1, 2, 3, 4, 5 )	2001-03-29	Chen Shuwen
( k = 0, 1, 2, 3, 4, 6 )	2017-05-26	Chen Shuwen
( k = 0, 1, 2, 4, 6, 8 )	2023-02-12	Chen Shuwen
( k = 0, 1, 2, 3, 4, 5, 6 )	2001-03-29	Chen Shuwen
( k = 0, 1, 2, 3, 4, 5, 6, 7 )	2023-04-08	Chen Shuwen
( k = -1, 0 )	2001-03-26	A.Moessner, Chen Shuwen
( k = -2, 0 )	2001-03-26	Bini, Dubouis and Mathieu, Chen Shuwen
( k = -3, 0 )	2001-05-08	Stephane Vandemergel, Chen Shuwen
( k = -4, 0 )	2001-05-08	Stephane Vandemergel, Chen Shuwen
( k = -2, -1, 0 )	2001-03-26	A.Golden, Chen Shuwen
( k = -3, -1, 0 )	2001-05-08	Chen Shuwen
( k = -4, -1, 0 )	2017-03-24	Chen Shuwen
( k = -3, -2, 0 )	2022-10-13	Chen Shuwen
( k = -4, -2, 0 )	2001-05-08	Chen Shuwen
( k = -5, -1, 0 )	2017-03-24	Chen Shuwen
( k = -6, -2, 0 )	2017-05-19	Chen Shuwen
( k = -3, -2, -1, 0 )	2001	Guo Xianqiang, Chen Shuwen
( k = -4, -2, -1, 0 )	2017-03-22	Chen Shuwen
( k = -6, -2, -1, 0 )	2017-05-31	Chen Shuwen
( k = -6, -4, -2, 0 )	2017-03-21	Chen Shuwen
( k = -4, -3, -2, -1, 0 )	2001	Guo Xianqiang, Chen Shuwen
( k = -5, -3, -2, -1, 0 )	2017-04-15	Chen Shuwen
( k = -6, -4, -2, -1, 0 )	2017-03-23	Chen Shuwen
( k = -5, -4, -3, -2, -1, 0 )	2001-03-29	Chen Shuwen
( k = -6, -4, -3, -2, -1, 0 )	2017-05-26	Chen Shuwen
( k = -8, -6, -4, -2, -1, 0 )	2023-02-12	Chen Shuwen
( k = -6, -5, -4, -3, -2, -1, 0 )	2001-03-29	Chen Shuwen
( k = -7, -6, -5, -4, -3, -2, -1, 0 )	2023-04-08	Chen Shuwen
( k = -1, 1 )	2001-05-06	Chen Shuwen
( k = -1, 2 )	2001-05-08	Chen Shuwen
( k = -2, 1 )	2001-05-08	Chen Shuwen
( k = -2, 2 )	2011	Ajai Choudhry
( k = -1, 0, 1 )	2001-05-08	Guo Xianqiang, Chen Shuwen
( k = -1, 0, 2 )	2017-03-25	Chen Shuwen
( k = -1, 1, 2 )	2017-03-20	Chen Shuwen
( k = -1, 1, 3 )	2017-03-20	Chen Shuwen
( k = -1, 1, 5 )	2017-04-07	Chen Shuwen
( k = -2, 0, 1 )	2017-03-25	Chen Shuwen
( k = -2, 0, 2 )	2014-09-06	Chen Shuwen
( k = -2, -1, 1 )	2017-03-20	Chen Shuwen
( k = -3, 0, 3 )	2017-03-25	Chen Shuwen
( k = -3, -1, 1 )	2017-03-20	Chen Shuwen
( k = -5, -1, 1 )	2017-04-07	Chen Shuwen
( k = -1, 0, 1, 2 )	2017-03-22	Chen Shuwen
( k = -1, 0, 1, 3 )	2017-03-22	Chen Shuwen
( k = -1, 1, 2, 3 )	2022-09-22	Chen Shuwen
( k = -2, 0, 2, 4 )	2017-05-31	Chen Shuwen
( k = -2, -1, 0, 1 )	2017-03-22	Chen Shuwen
( k = -2, -1, 1, 2 )	2017-03-30	Chen Shuwen
( k = -3, -1, 0, 1 )	2017-03-22	Chen Shuwen
( k = -3, -2, -1, 1 )	2022-09-22	Chen Shuwen
( k = -4, -2, 0, 2 )	2017-05-31	Chen Shuwen
( k = -1, 0, 1, 2, 3 )	2017-03-22	Chen Shuwen
( k = -1, 0, 1, 2, 4 )	2017-05-26	Chen Shuwen
( k = -2, -1, 0, 1, 2 )	2017-03-29	Chen Shuwen
( k = -3, -1, 0, 1, 3 )	2017-03-30	Chen Shuwen
( k = -3, -2, -1, 0, 1 )	2017-03-22	Chen Shuwen
( k = -4, -2, -1, 0, 1 )	2017-05-26	Chen Shuwen
( k = -1, 0, 1, 2, 3, 4 )	2017-05-21	Chen Shuwen
( k = -4, -3, -2, -1, 0, 1 )	2017-05-21	Chen Shuwen
( k = -3, -2, -1, 0, 1, 2, 3 )	2017-03-30	Chen Shuwen

 

• Ideal integer solutions ( T/L = 65 types )

Type	First solution found in	First solution found by
( h = 1, 2, 4 )
( h = 1, 2, 6 )	1997-05-11	Chen Shuwen
( h = 1, 3, 4 )	1991	Ajai Choudhry
( h = 2, 3, 4 )	2001	Ajai Choudhry
( h = 1, 2, 3, 5 )	1944	A.Golden
( h = 1, 2, 4, 6 )	1953	G.Palama
( h = 1, 2, 3, 4, 6 )	1944	A.Golden
( h = 1, 2, 3, 5, 7 )	1953	G.Palama
( h = 1, 2, 4, 6, 8 )	1940	A.Letac
( h = 1, 2, 3, 4, 5, 7 )	1944	A.Golden
( h = 1, 2, 3, 4, 6, 8 )	2019-08-01	Jaroslaw Wroblewski, Chen Shuwen
( h = 1, 2, 3, 5, 7, 9 )	2000-10-16	Chen Shuwen
( h = 1, 2, 4, 6, 8, 10 )	1999-09-08	Nuutti Kuosa, Jean-Charles Meyrignac, Chen Shuwen
( h = 1, 2, 3, 4, 5, 6, 8 )	1997-05-19	Chen Shuwen
( h = 1, 2, 3, 4, 5, 6, 7, 9 )	1997-05-19	Chen Shuwen
( h = -4, -2, -1 )	2001-03-26	Chen Shuwen
( h = -6, -2, -1 )	2001-03-26	Chen Shuwen
( h = -4, -3, -1 )	2001-03-26	Ajai Choudhry, Chen Shuwen
( h = -4, -3, -2 )	2001-03-26	Ajai Choudhry, Chen Shuwen
( h = -5, -3, -2, -1 )	2001-03-26	A.Golden, Chen Shuwen
( h = -6, -4, -2, -1 )	2001-03-26	G.Palama, Chen Shuwen
( h = -6, -4, -3, -2, -1 )	2001-03-26	A.Golden, Chen Shuwen
( h = -7, -5, -3, -2, -1 )	2001-03-26	G.Palama, Chen Shuwen
( h = -8, -6, -4, -2, -1 )	2001-03-26	A.Letac, Chen Shuwen
( h = -7, -5, -4, -3, -2, -1 )	2001-03-26	A.Golden, Chen Shuwen
( h = -8, -6, -4, -3, -2, -1 )	2019-08-01	Jaroslaw Wroblewski, Chen Shuwen
( h = -9, -7, -5, -3, -2, -1 )	2001-03-26	Chen Shuwen
( h = -10, -8, -6, -4, -2, -1 )	2001-03-26	Nuutti Kuosa, Jean-Charles Meyrignac, Chen Shuwen
( h = -8, -6, -5, -4, -3, -2, -1 )	2001-03-26	Chen Shuwen
( h = -9, -7, -6, -5, -4, -3, -3, -2, -1 )	2001-03-26	Chen Shuwen
( h = 0, 1, 3 )	1939	A.Moessner
( h = 0, 1, 2, 4 )	2011	Ajai Choudhry
( h = 0, 1, 3, 5 )	2017-03-26	Chen Shuwen
( h = 0, 1, 2, 3, 5 )	2011	Ajai Choudhry
( h = 0, 1, 2, 4, 6 )	2017-03-21	Chen Shuwen
( h = 0, 1, 2, 3, 4, 6 )	2011	Ajai Choudhry
( h = -3, -1, 0 )	2001-03-26	A.Moessner, Chen Shuwen
( h = -4, -2, -1, 0 )	2011	Ajai Choudhry, Chen Shuwen
( h = -5, -3, -1, 0 )	2017-03-26	Chen Shuwen
( h = -5, -3, -2, -1, 0 )	2011	Ajai Choudhry, Chen Shuwen
( h = -6, -4, -2, -1, 0 )	2017-03-21	Chen Shuwen
( h = -6, -4, -3, -2, -1, 0 )	2011	Ajai Choudhry, Chen Shuwen
( h = -1, 0, 2 )	2017-03-26	Chen Shuwen
( h = -1, 1, 2 )	2011	Ajai Choudhry
( h = -2, 0, 1 )	2017-03-26	Chen Shuwen
( h = -2, -1, 1	2011	Ajai Choudhry, Chen Shuwen
( h = -1, 0, 1, 3 )	2017-03-26	Chen Shuwen
( h = -1, 0, 1, 5 )	2017-04-07	Chen Shuwen
( h = -1, 0, 2, 4 )	2019-02-11	Chen Shuwen
( h = -1, 1, 2, 3 )	2011	Ajai Choudhry
( h = -2, 0, 1, 2 )	2017-03-25	Chen Shuwen
( h = -2, -1, 0, 2 )	2017-03-25	Chen Shuwen
( h = -2, -1, 1, 2 )	2011	Ajai Choudhry
( h = -2, -1, 1, 3 )	2023-05-12	Chen Shuwen
( h = -3, -1, 0, 1 )	2017-03-26	Chen Shuwen
( h = -3, -1, 1, 2 )	2023-05-12	Chen Shuwen
( h = -3, -2, -1, 1 )	2011	Ajai Choudhry, Chen Shuwen
( h = -4, -2, 0, 1 )	2019-02-11	Chen Shuwen
( h = -5, -1, 0, 1 )	2017-04-07	Chen Shuwen
( h = -1, 0, 1, 2, 4 )	2019-02-05	Chen Shuwen
( h = -4, -2, -1, 0, 1 )	2019-02-05	Chen Shuwen
( h = -1, 0, 1, 2, 3, 5 )	2017-03-27	Chen Shuwen
( h = -1, 1, 2, 3, 4, 5 )	2011	Ajai Choudhry
( h = -5, -3, -2, -1, 0, 1 )	2017-03-27	Chen Shuwen
( h = -5, -4, -3, -2, -1, 1 )	2011	Ajai Choudhry, Chen Shuwen

 

• Ideal prime solutions ( T/L = 24 types )

Type	First solution found in	First solution found by
( k = 1 )
( k = 2 )	2000-09-08	Jean-Charles Meyrignac
( k = 3 )	2000-09-08	Jean-Charles Meyrignac
( k = 4 )	2000-09-03	Jean-Charles Meyrignac
( k = 1, 2 )	1964	Albert H. Beiler
( k = 1, 3 )	2016-09-26	Chen Shuwen
( k = 1, 4 )	2016-09-25	Chen Shuwen
( k = 1, 5 )	2002-01-03	Jaroslaw Wroblewski
( k = 2, 4 )	2016-09-25	Chen Shuwen
( k = 2, 6 )	2016-10-17	Aloril, Jean-Charles Meyrignac, Chen Shuwen
( k = 1, 2, 3 )	1964	Albert H. Beiler
( k = 1, 2, 4 )	2017-03-12	Chen Shuwen
( k = 1, 3, 5 )	2016-09-25	Torbjorn Alm, Jean-Charles Meyrignac, Chen Shuwen
( k = 2, 4, 6 )	2017-03-14	Chen Shuwen
( k = 1, 2, 3, 4 )	2016-09-26	Chen Shuwen
( k = 1, 2, 3, 5 )	2016-03-12	Chen Shuwen
( k = 1, 2, 4, 6 )	2017-03-15	Chen Shuwen
( k = 1, 3, 5, 7 )	1999	Carlos Rivera
( k = 1, 2, 3, 4, 5 )	1999-08-31	T.W.A. Baumann
( k = 1, 2, 3, 4, 5, 6)	2016	Qiu Min
( k = 1, 2, 3, 4, 5, 6, 7 )	1999-09-01	T.W.A. Baumann
( k = 1, 2, 3, 4, 5, 6, 7, 8 )	2016-09-30	Chen Shuwen
( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 )	2016-10-04	Chen Shuwen
( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 )	2023	Jaroslaw Wroblewski