Equal Sums of Like Powers

What is New on This Site

• Algebraic Identities ( The Generalization of Ramanujan's 6-10-8 Identity )
  • Identities for ( k = 0, 1 ) , ( k = 0, 1, 2 )

• Algebraic Identities ( The Generalization of Ramanujan's 6-10-8 Identity )
  • Identities for ( k = 1, 2 ) , ( k = 1, 3 ) , ( k = 1, 4 ) and ( k = 1, 5 )

• Algebraic Identities ( The Generalization of Ramanujan's 6-10-8 Identity )
  • New page on algebraic identities

• Ideal positive integer solutions (k<0) / Equal sums of like powers / Equal products and equal sums of like powers
• Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = -2, 0, 2, 4 ), ( k = -4, -2, 0, 2 ), ( k = 0, 1, 2, 6 ), ( k = -6, -2, -1, 0 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 21 + 20 + 31 = 150 types are solved.

• Ideal positive integer solutions (k<0) / Equal sums of like powers / Equal products and equal sums of like powers
• Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = -1, 0, 1, 2, 4 ), ( k = -4, -2, -1, 0, 1 ), ( k = 0, 1, 2, 3, 4, 6 ), ( k = -6, -4, -3, -2, -1, 0 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 20 + 19 + 29 = 146 types are solved.

• Ideal positive integer solutions (k<0) / Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = -1, 0, 1, 2, 3, 4 ) and ( k = -4, -3, -2, -1, 0, 1 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 19 + 18 + 27 = 142 types are solved.

• Equal products and equal sums of like powers / Ideal positive integer solutions (k<0)
• Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = 0, 2, 6 ), ( k = -6, -2, 0 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 19 + 18 + 25 = 140 types are solved.

• ( h = 1, 3, 4 )
  • 5-7-2-6 identity by Chen Shuwen.

• Equal products and equal sums of like powers / Ideal positive integer solutions (k<0)
• Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = 0, 1, 2, 3, 5 ), ( k = -5, -3, -2, -1, 0 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 18 + 17 + 25 = 138 types are solved.
• Equal sums of like powers / Records on equal sums of like powers
  • The first prime solution founders of ( k = 1, 2 ), ( k = 1, 2, 3 ) are comfirmed. (Thanks for Carlos Rivera's help.)
  • The date for first prime solution of ( k = 1, 3, 5, 7 ) are comfirmed.(Thanks Carlos Rivera.)

• Equal products and equal sums of like powers / Records on equal sums of like powers
  • The first non-negative integer solution founders of ( k = 0, 2 ), ( k = 0, 3 ), ( k = 0, 4 ) are corrected.
• Equal sums of like powers / Records on equal sums of like powers
  • The first prime solution founders of ( k = 1, 2 ), ( k = 1, 2, 3 ), ( k = 1, 2, 3, 4, 5 ), ( k = 1, 3, 5, 7 )( k = 1, 2, 3, 4, 5, 6, 7 ) are corrected.
• Ideal integer solutions (h<0)
  • Parametric ideal integer solution of ( h = -1, 1, 2, 3 ) and ( h = -1, 1, 2, 3, 4, 5 ) by A.Choudhry in 2011.
  • Parametric ideal integer solution of ( h = -1, 0, 1, 3 ) and ( h = -1, 0, 1, 2, 3, 5 ) by Chen Shuwen in 2017.

• Ideal positive integer solutions (k<0) / Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = -2, -1, 0, 1, 2 ), ( k = -3, -1, 0, 1, 3 ), ( k = -2, -1, 1, 2 ) and ( k = -3, -2, -1, 0, 1, 2, 3 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 17 + 16 + 23 = 134 types are solved.

• Equal products and equal sums of like powers / Ideal integer solutions (h<0)
• Equal sums of like powers / Records on equal sums of like powers
  • Ideal integer solutions for ( h = -1, 0, 1, 3 ), ( h = -3, -1, 0, 1 ), ( h = -1, 0, 2 ), ( h = -2, 0, 1 ), ( h = -1, 0, 1, 2, 3, 5 ), ( h = -5, -3, -2, -1, 0, 1 ) are found by Chen Shuwen.
    • Ideal integer solutions: 14 + 14 + 6 + 6 + 15 = 55 types are solved.

• Equal products and equal sums of like powers / Ideal integer solutions (h<0)
• Equal sums of like powers / Records on equal sums of like powers
  • Ideal integer solutions for ( h = 0, 1, 3, 5 ), ( h = -5, -3, -1, 0 ) are found by Chen Shuwen.
    • Ideal integer solutions: 14 + 14 + 6 + 6 + 9 = 49 types are solved.

• Ideal positive integer solutions (k<0) / Ideal integer solutions (h<0)
• Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = -1, 0, 2 ), ( k = -2, 0, 1 ), ( k = -3, 0, 3 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 17 + 16 + 19 = 130 types are solved.
  • Ideal integer solutions for ( h = -2, 0, 1, 2 ), ( h = -2, -1, 0, 2 ) are found by Chen Shuwen.
    • Ideal integer solutions: 14 + 14 + 5 + 5 + 9 = 47 types are solved.

• Equal products and equal sums of like powers / Ideal positive integer solutions (k<0)
• Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = 0, 1, 4 ), ( k = 0, 1, 5 ), ( k = -4, -1, 0 ), ( k = -5, -1, 0 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 17 + 16 + 16 = 127 types are solved.

• Equal products and equal sums of like powers / Ideal positive integer solutions (k<0)
• Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = 0, 1, 2, 4, 6 ), ( k = -6, -4, -2, -1, 0 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 15 + 14 + 16 = 123 types are solved.

• Ideal positive integer solutions (k<0) / Equal sums of like powers / Equal products and equal sums of like powers
• Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = -1, 0, 1, 2 ), ( k = -1, 0, 1, 3 ), ( k = -1, 0, 1, 2, 3 ),( k = -2, -1, 0, 1 ), ( k = -3, -1, 0,1 ) , ( k = -3, -2, -1, 0,1 ), ( k = 0, 1, 2, 4 ), ( k = -4, -2, -1, ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 14 + 13 + 16 = 121 types are solved.

• Equal products and equal sums of like powers / Ideal positive integer solutions (k<0) / Ideal integer solutions (h<0)
• Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = 0, 2, 4, 6 ), ( k = -6, -4, -2, 0 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 13 + 12 + 10 = 113 types are solved.
  • Ideal integer solutions for ( h = 0, 1, 2, 4, 6 ), ( h = -6, -4, -2, -1, 0 ) are found by Chen Shuwen.
    • Ideal integer solutions: 14 + 14 + 5 + 5 + 7 = 45 types are solved.

• Ideal positive integer solutions (k<0) / Equal sums of like powers / Records on equal sums of like powers
  • Ideal non-negative integer solutions for ( k = -1, 1, 2 ), ( k = -1, 1, 3 ), ( k = -2, -1, 1 ), ( k = -3, -1, 1 ) are found by Chen Shuwen.
    • Ideal non-negative integer solutions: 39 + 39 + 12 + 11 + 10 = 111 types are solved.

• Equal sums of like powers / Records on equal sums of like powers
  • Prime solution for ( k = 1, 2, 4 ), ( k = 2, 4, 6 ), ( k = 1, 2, 3, 5 ), ( k = 1, 2, 4, 6 ) are found by Chen Shuwen.
  • Ideal prime solutions: 23 types are solved.
  • The first prime solution founders of ( k = 1, 2, 3 ), ( k = 1, 2, 3, 4 ), ( k = 1, 2, 3, 4, 5, 6 ) are corrected. ( Qiu Min and Wu Qiang found prime solutions of these three types before April 2016, just five months earlier than Chen Shuwen.)
• ( k = 1, 2, 3, 4, 6 )
  • Parametric ideal non-negative integer solution of ( k = 0, 1, 2, 3, 4 ) is found, by A.Choudhry in 2016.
  • Parametric ideal non-negative integer solution of ( k = 1, 2, 3, 4, 6 ) is found, by Chen Shuwen in 2017.
• ( k = 1, 2, 3, 4, 5, 6, 7, 8 )
• There is no new new symmetric solution in the range of max { a_i, b_i }< 13000. (By Chen Shuwen in 2016)
• There is no non-symmetric solution in the range of max { a_i, b_i }< 310. (By Chen Shuwen in 2016-2017)
• ( k = 1, 3, 5, 7, 9 )
  • There is no new positive integer solution in the range of Max{a_i,b_i}<=5600.
  • There is no ideal symmetric non-negative solutions of ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ) in the range of Max{a_i,b_i} <= 50000.
• ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 )
  • Only one prime solution is found in the range of of { a_i, b_i }< 100000000, by Chen Shuwen in 2016.
• ( k = 2, 4, 6, 8, 10 )
  • There is no new solution in the range of of max {a_i, b_i } <= 1800.
• Equal products and equal sums of like powers
  • Smallest solution for ( k = 0, 1, 2, 3, 4, 5, 6 )

• Equal sums of like powers / Records on equal sums of like powers
  • Prime solution for ( k = 2, 6 ) by Aloril, Jean-Charles Meyrignac, Chen Shuwen
  • Ideal prime solutions: 19 types are solved.
  • The first prime solution founders of ( k = 2 ), ( k = 3 ), ( k = 4 ), ( k = 1, 5 ), ( k = 1, 3, 5 ) and ( k = 1, 3, 5, 7 ) are corrected. 

• New website is distributed: eslpower.org
• Equal sums of like powers
  • Ideal non-negative integer solutions: 39 + 39 + 12 + 11 + 6 = 107 types are solved.
  • Ideal integer solutions: 14 + 14 + 4 + 4 + 7 = 43 types are solved.
  • Ideal prime solutions: 18 types are solved.
• The Prouhet-Tarry-Escott problem
  • Prime solutions for all the types n<=9
• Equal products and equal sums of like powers updated
• Ideal positive integer solutions (k<0) updated
• Ideal integer solutions (h<0) new page added
• Records on equal sums of like powers three lists updated

• First solution for ( h = 2, 3, 4 ) by Ajai Choudhry.
• Equal sums of like powers
• Records on equal sums of like powers
• T/L = 39 types ( m = n + 1, k > 0, non-negative integers)
• T/L = 14 types ( m = n, h > 0, integers)
• Links and References : [64] is added.
• Multigrade Chains : some solutions for ( k = 1, 3 ) by Jarek Wroblewski.

• Equal sums of like powers
• T/L = 105 types ( m = n + 1, non-negative integers)
• Results and discussion on the k < 0 case.
• First known solution of ( k = -1, 2 ) , ( k = -2, 1 ) , ( k = -1, 0, 1 )
• T/L = 4 types ( m = n + 1, k1 < 0, kn >0, non-negative integers)
• Equal Products and Equal sums of like powers
• First known solution of ( k = 0, 3 ) , ( k = 0, 4 ) , ( k = 0, 1, 3 ) , ( k = 0, 2, 4 )
• T/L = 12 types ( m = n + 1, k1 = 0, others k >0, non-negative integers)

• Equal sums of like powers
• T/L = 94 types ( m = n + 1, non-negative integers)
• Unsolved problems and conjectures
• Question I by Chen Shuwen and Conjectures by Chen Shuwen are modified.
• A new page is added: Multigrade Chains.
• T/L = 10 types ( m = n + 1, j>=3, non-negative integers)
• T/L = 7 types ( m = n + 1, any j, non-negative integers)
• T/L = 2 types ( m = n , j>=3, integers)
• A new page is added: Results and discussion on the k < 0 case.
• First known solution of ( k = -1, 1).
• A new page is added: ( k = -1, 1).

• Equal sums of like powers
• Only consider k > 0 in the page.
• Records on equal sums of like powers
• T/L = 39 types ( m = n + 1, k > 0, non-negative integers)
• T/L = 13 types ( m = n, h > 0, integers)
• Equal Products and Equal sums of like powers
• Frist solution for ( k = 0, 1, 2, 3 ) by Quo Xianqiang.

• The main site is moved to: http://go6.163.com/~eslp (Redirect URL: http://eslp.yeah.net )
• A new page is added: Equal Products and Equal sums of like powers
• First solution for ( k = 0 ), ( k = 0, 1 ), ( k = 0, 1 ), ( k = 0, 2 ), ( k = 0, 1, 2 ) and ( h = 0, 1, 3 )
• Frist solution for ( k = 0, 1, 2, 3 ), ( k = 0, 1, 2, 3, 4 ), ( k = 0, 1, 2, 3, 4, 5 ), ( k = 0, 1, 2, 3, 4, 5, 6 )
• Records on equal sums of like powers
• T/L =47 types (m=n+1, non-negative integers)
• T/L =14 types (m=n, integers)
• Equal sums of like powers
• New solution for ( h = 1, 2, 6 ) by Ajai Choudhry
• All the pages have been modified.

• First solution for ( k = 1, 3, 7 ) by Ajai Choudhry in 1999.
• Equal sums of like powers
• Thanks for Jean-Charles Meyrignac mirror this site: http://euler.free.fr/eslp
• Records on equal sums of like powers
• T/L =39 types (m=n+1, non-negative integers)
• T/L =13 types (m=n, integers)

• First solution for ( k = 1, 3, 5, 7, 9 ) by Chen Shuwen
• First solution for ( h = 1, 2, 3, 5, 7, 9 ) by Chen Shuwen
• New solution for ( k = 2, 4, 6, 8 )
• New solution for ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9 )
• Records on equal sums of like powers
• T/L =39 types (m=n+1, non-negative integers)
• T/L =13 types (m=n, integers)

• First solution for ( h = 1, 3, 4 ) by Ajai Choudhry
• Records on equal sums of like powers
• T/L =38 types (m=n+1, non-negative integers)
• T/L =12 types (m=n, integers)

• First solution for ( k = 1, 3, 7 )
• Records on equal sums of like powers
• T/L =38 types (m=n+1, non-negative integers)
• T/L =11 types (m=n, integers)

• First solution for ( h = 1, 2, 4, 6, 8, 10 )
• Records on equal sums of like powers
• T/L =37 types (m=n+1, non-negative integers)
• T/L =11 types (m=n, integers)

• First solution for ( k = 2, 4, 6, 8, 10 )
• First solution for ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 )
• Records on equal sums of like powers
• T/L =37 types.
• The Prouhet-Tarry-Escott problem
• Great progress on ( k = 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 )
• Unsolved problems and conjectures

• First solution for ( k = 1, 2, 3, 5, 7 ) 
• Records on equal sums of like powers
• T/L =35 types.
• Non-negative integer solutions of the m=n+2 case
• ( k = 2, 4, 8 ) by Randy L. Ekl
• Other results on equal sums of like powers
• A link to Jean-Charles Meyrignac's site
• Discussion and comments
• Some E-mails about Equal sums of likes recently.

• First solution for ( k = 1, 2, 3, 6 ) 
• First solution for ( k = 1, 2, 3, 4, 5, 6, 8 ) 
• Records on equal sums of like powers
• New solutions on ( k = 1, 2, 3, 4, 5, 7 ) 
• Internet resources collection

• Internet resources collection
• Discussion and comments

• Chen's Equal sums of like powers site is moved to http://www.nease.net/~chin/eslp.
• Discussion and comments
• Some E-mails sent to Chen during Auget 1 to December 31, 1997.
• A new solution of ( 4, 1, 3 ) by Allan MacLeod.
• Sign gusetbook and View Guestbook
• Internet resources collection
• Wide Area Distributed Computation: The Prouhet-Tarry-Escott Problem

• Internet resources collection
• Sums of square roots ( An application of the Prouhet-Tarry-Escott problem ).
• Unsolved problems and conjectures
• P.Borwein and C.Ingalls had found and proved the ( k = 1, 2, ..., n ) case of Conjecture II in 1994.
• Discussion and comments
• Some E-mails sent to Chen during July 13 to Auget 1.
• References
• [59] is added.

• Other results on equal sums of like powers
• The first solution of the type ( 5, 2, 3 ) is known.
• Internet resources collection
• Some new resources are added to this collection page.
• Unsolved problems and conjectures
• The first version of this page is finished.
• Discussion and comments
• E-mails sent to Chen discussing the topic " Equal sums of like powers" or commenting on Chen's site.
• References
• [48]~[58] and "Other References" are added.
• What is new
• This page is added for the first time.
• ( k = 1, 4 )
• Chen Shuwen obtained a parametric solution of this type in July 1997.

• Chen Shuwen's Euqal Sums of Like Powers site is published.