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• Computing Minimal Equal Sums Of Like Powers
  • Jean-Charles Meyrignac's distributed-computing project on equal sums of like powers and the place to look for the current status of the problem.
  • The Links page is also helpful: http://euler.free.fr/links.htm
• Internet Resources Collection on Equal Sums of Like Powers
  • By Chen Shuwen, last revised July 14, 1999.
• The Prouhet-Tarry-Escott Problem Revisited
  • A good online reference, by Peter Borwein and Colin Ingalls.
• Science/Math/Number_Theory/Diophantine_Equations/Equal_Sums_of_Like_Powers

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• [1] Hua L.K., Introduction to Number Theory, Springer, 1982.
• [2] H.L.Dorwart and O.E.Brown, The Tarry-Escott Problem, Amer. Math. Monthly, 44(1937), 613-626.
• [3] G.H.Hardy and E.M.Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, London, 1979, 328-339.
• [4] E.Rees and C.Smyth, On the constant in the Tarry-Escott problem, Fifity Years of Poly- nomials, Lecture Notes in Math., vol. 1415, Springer, Berlin and New York, 1990, 196-208 MR91g:11030.
• [5] A.Gloden, Mehrgradige Gleichungen, Noordhoff, Groningen, 1944.
• [6] T.N.Sinha, Diophantine systems of the Tarry-Escott type, Anupam ,India,1984
• [7] J.Chernick, Ideal solutions of the Tarry-Escott problem, Amer. Math. Monthly, 44(1937), 626-633
• [8] H.Gupta, A system of equations having no nontrivial solutions, J. of Research, National Bureau of Standards 71B(1967), 181-182
• [9] L.E.Dickson, History of the theory of numbers, vol. II: Diophantine Analysis, reprint, New York: Chelsea, 1966, 550-552, 705-716
• [10] L.J.Lander and T.R.Parkin, Equal sums of biquadrates, Math. Comp., 20(1966), 450-451
• [11] L.Lander, T.Parkin and J.Selfridge, A survey of equal sums of like powers, Math. Comp., 21(1967), 446-459
• [12] A.J.Zajta, Solutions of the Diophantine Equation A⁴+B⁴=C⁴+D⁴, Math. Comp., 41(1983), 635-659
• [13] A.Moessner, Einige Numerische Identitaten, Proc. Indian Acad. Sci. Sect. A, 10(1939), 296-306
• [14] L. J. Lander, Geometric Aspects of Diophantine Equations Involving Equal Sums of Like Powers, Amer. Math. Monthly, 75(1968), 1061-1073
• [15] J.Delorme, On the Diophantine Equation x₁⁶+ x₂⁶+ x₃⁶ = y₁⁶+ y₂⁶+ y₃⁶, Math. Comp., 59(1992), 703-715
• [16] R.N.Singh, Equal sum of like powers, Math. Student, 56(1988), no,1-4, 192-194, MR90m:11044.
• [17] G.Xeroudakes and A.Moessner, On equal sums of like powers, Proc. Indian Acad. Sci. Sect.A, 48(1958), 245-255
• [18] A.Choudhry, The Diophantine system Sum(x_i^r)=Sum(y_i^r), r =1,3,5., Bull. Calcutta Math. Soc. 83(1991), 85-86
• [19] G.Palama, Diophantine systems of the type Sum(a_i^k)=Sum(b_i^k),( k=1, 2, ..., n, n+2, n+4,..., n+2r ), Scripta Math., 19(1953), 132-134
• [20] A.Gloden, Two theorems on multi-degree equalities, Amer. Math. Monthly, 55(1948), 86-88
• [21] A.Moessner, On equal sums of powers, Math. Student, 15(1947), 83-88
• [22] C.J.Smyth, Ideal 9th-Order Multigrades and Letac's Elliptic Curve, Math. Comp., 57(1991), 817-823, MR92b:11019
• [23] G.Tarry, L'intermediaire des mathematiciens,19(1912), 200, 219-221; 20(1913), 68-70.
• [24] E.B.Escott, Logarithmic Series, Quarterly Journal of Math., 41(1910), 141-167.
• [25] A.Letac, Gazeta Matemetica, 48(1942), 68-69
• [26] Simcha Brundo and Irving Kaplansky, Equal sums of sixth powers, J.Number Theory, 6(1974),401-403
• [27] Simcha Brundo, Triples of sixth powers with equal sums, Math. Comp. 30(1976),646-648
• [28] Andrew Bremner, A geometric approach to equal sums of sixth powers, Proc. London Math. Soc. 43(1981), 544-581
• [29] Simcha Brundo, On generating infinitely many solutions of the Diophantine equation A⁶+B⁶+C⁶ = D⁶+E⁶+F⁶, Math. Comp. 24(1970), 453-454.
• [30] Simcha Brundo, Some new results on Equal sums of like powers, Math. Comp. (23)1969,877-880.
• [31] K.Subba Rao, On sums of sixth powers, J.London Math. Soc.,v.9, 1934, 172-173.
• [32] T.N.Sinha, A note on equal sums of like powers, Math. Student, 46(1978), no.2-4, 121-123, MR84e:10025
• [33] G.Palama, Multigrade normali del 9^o ordine inverso del teorema di Gloden, Uni. Roma 1st Naz. Alta Mat. Rend. Mat e Appl., 1950,Vol.5,Part.9, 229-236.
• [34] T.N.Sinha, Some systems of Diophantine equations of the Tarry-Escott type, Jour. Ind. Math. Soc. V.30,1966,15-25.
• [35] A.Gloden, Parametric solutions of two multi-degreed equalities, Amer. Math. Monthly, v.55, 1948, 86-88
• [36] A.Gloden, Uber mehrgradige Gleichungen, Arch. Math., v.1, 1949, 482-483
• [37] A.Golden, Note on systems of diophantine equations, Scrip.Math., 15(1949), 163-164.
• [38] A.Gloden, Zwei Parameterlosungen einer mehrgradigen Gleichung, Arch.Math., v.1, 1949, 480-482
• [39] J.L.Burchnall & T.W.Chaundy, A type of "Magic Square" in Tarry's problem,Quart. J. Math., 8(1937), 119-130
• [40] A. Choudhry, Multigrade Chains with Odd Exponents, J.Number Theory, 42(1992), 134-140, MR94a:11044
• [41] J.H.Silverman, Taxicabs and sums of two cubes, Amer. Math.Monthly,1993, 331-340.
• [42] E.Rosenstiel, J.A.Dardis and C.R.Rosenstiel, The four least solutions in distinct positive integers of the Diophantine equation s=x³+y³=z³+w³=u³+v³=m³+n³, Bull. Inst. Math. Appl. 27(1991), no.7, 155-157, MR92i:11134.
• [43] J.Leech, Some solutions of Diophantine equations. Proc. Cambridge Philos. Soc.,v.53,(1957), 778-780.
• [44] Peter Borwein and Colin Ingalls, The Prouhet-Tarry-Escott Problem Revisited, Enseign.Math. (2).40.(1994),no.1-2,3-27.
• [45] Noam D. Elkies, On A⁴ = B⁴ + C⁴ + D⁴, Math. Comp. (51)1988, 825-835
• [46] L.J.Lander and T.R.Parkin, A counterexamples to Euler's sums of powers conjecture, Math. Comp., 21(1967), 101-103
• [47] Randy L. Ekl, Equal sums of four seventh powers, Math. Comp., 65 (1996), pp. 1755-1756.
• [48] T.N.Sinha, Some systems of Diophantine equations of the Tarry-Escott type, Jour. Ind. Math. Soc., 30(1966), 15-25.
• [49] Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.
• [50] L.Hahn, The Tarry-Escott Problem (Problem 10284), Amer. Math. Monthly,1995, 843-844.
• [51] H.L.Dorwart, Sequences of ideal solutions in the Tarry-Escott problem, Bull. Amer. Math. Soc., 53(1947), 381-391.
• [52] S.Sastry, On equal sums of like powers, Math. Student, 15(1947), 29-32.
• [53] A.Moessner, On the multiple identity x₁ⁿ + x₂ⁿ + x₃ⁿ + x₄ⁿ + x₅ⁿ = y₁ⁿ + y₂ⁿ + y₃ⁿ + y₄ⁿ + y₅ⁿ for n =1, 3, 5, 7 , Scrip. Math. 18(1952), 90-91.
• [54] A.Bremner and R.K.Guy, A Dozen Difficult Diophantine Dilemmas, Amer. Math. Monthly, 1988, 31-36.
• [55] S.Sastry and T.Rai, On equal sums of like powers, Math. Student, 16(1948), 18-19.
• [56] S.Sastry, On sums of powers, Jour.London.Math.Soc., 9(1934), 170-171.
• [57] J.Leech, On A⁴ + B⁴ + C⁴ + D⁴= E⁴, Proc. Cambridge Philos .Soc., 54(1958), 554-555.
• [58] T.N.Sinha, On the Tarry-Escott problem, Amer. Math. Monthly, 7391966), 280-285.
• [59] A.Choudhry, Symmetric Diophantine systems, Acta Arithmetica, 59(1991), 291-307.
• [60] A.Choudhry, Equal sums of Seventh powers, Rocky Mountain Journal of Mathematics, Volume 30, Number 3, Fall 2000
• [61] A.Choudhry, On Equal sums of Sixth powers, Rocky Mountain Journal of Mathematics, Volume 30, Number 3, Fall 2000
• [62] A.Choudhry, Idea Solutions of the Tarry-Escott problem of Degree Four and a Related Diophantine System, L'Enseignement Mathematique, Vol. 46 (2000), pp.313-323.
• [63] Randy L. Ekl, New results in equal sums of like powers, Math. Comp. 67 (1998), 1309-1315.
• [64] A.Choudhry, Equal sums of like powers, Rocky Mountain Journal of Mathematics, Volume 31, Number 1, 2001

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The following are some references on "Equal sums of like powers" or "The Prouhet-Tarry-Escott Problem" which are useful but have not yet been quoted in Chen Shuwen's Equal sums of like powers pages.

• D.H.Lehmer, The Tarry-Escott problem, Scripta Math, 1947, 37-41.
• E.M.Wright, The Prouhet-Lehmer problem, Jour.London.Math.Soc, 23(1948), 279-285.
• Hua Loo-Keng, On Tarry's problem, Quart. J. of Math, 9(1938), 315-320.
• L.J.Lander, Three thirteens, Math. Comp., 27(1973), 397.
• E.M.Wright, On Tarry's problem (I), Quart. J. of Math, 6(1935), 261-267.
• E.M.Wright, On Tarry's problem (II), Quart. J. of Math, , 7(1936), 43-45.
• E.M.Wright, On Tarry's problem (III), Quart. J. of Math, , 8(1937), 48-50.
• H.Gupta, Selected Topics in Number Theory, Abacus Press, Kent, 1980, 322-331.
• E.M.Wright, Equal sums of like powers, Bull. Amer. Math. Soc., 54(1948), 755-757.
• E.M.Wright, Equal sums of like powers, Proc. Edin. Math. Soc., 8(1949), 138-142.
• Hua Loo-Keng, Improvement of a result of Wright, Jour.London.Math.Soc, 24(1949), 157-159.
• E.M.Wright, An 'easier' Waring problems, Jour.London.Math.Soc, 9(1934), 267-272.
• E.M.Wright, The Tarry-Escott and the 'easier' Waring problems, J.Reine Angew.Math, 311/312(1979), 170-173.
• I.Barrodale, A note on equal sums of like powers, Math. Comp., 20(1966), 318-322.
• E.M.Wright, Prouhet's 1851 solution of the Tarry-Escott problem of 1910, Amer. Math. Monthly,66(1959), 199-201.
• H.Kleiman, A note on the Tarry-Escott problem, J.Reine Angew. Math., 278/279(1975), 48-51.