Computational Number Theory
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[144. Four positive numbers such that the sum of any two is a cube (2013.5.15)]
[143. Solutions of x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4 (2013.5.1)]
[142. Solutions of x^4 + ay^4 = z^4 + bt^4 (2013.4.1)]
[141. Solutions of x^2 + y^4 = 5z^4 (2013.3.1)]
[140. A parametric solution of X^k + Y^k + Z^k + W^k + S^k = {U^2,V^4} for k=2,4 (2013.2.1)]
[139. Happy New Year! (2013.1.1)]
[138. Equal Sums of Seventh Powers (2012.11.30)]
[137. Many solutions of n=x^3 + y^3 + 2z^3 (2012.10.18)(2012.9.24)]
[136. Solutions of ax^6 + by^3 = cz^2 (2012.9.10)]
[135. Equal Sums of Eighth Powers Part 4 (2012.8.10)]
[134. New solution: 4388 = -18105946705^3 + 18108094211^3 - 2*1018372219^3 (2012.7.12)]
[133. Solutions of x^4 + y^4 + z^4 = w^2 (2012.7.2)]
[132. Equal Sums of Eighth Powers Part 3 (2012.6.11)]
[131. A collection of the parametric solutions for equal sums of sixth powers (2012.5.3)]
[130. A collection of the parametric solutions for equal sums of fifth powers (2012.4.2)]
[129. Equal Sums of Fifth Powers (2012.3.2)]
[128. Equal sums and equal products (2012.2.1)]
[127. Happy New Year! (2012.1.3)]
[126. Any non-zero rational number m is the sum of three rational cubes
in an infinite number of non-trivial ways. (2011.12.6)]
[125. New parametric solutions of x^3 + y^3 + z^3 = t^2 (2011.11.16)]
[124. Equal Sums of Eighth Powers et 1 (2011.10.11)]
[123. Equal Sums of Fourth Powers et 1 (2011.9.13)]
[122. Equal Sums of Fifth Powers et 1 (2011.8.23)]
[121. Equal Sums of Sixth Powers et 1 (2011.8.1)]
[120. Equal Sums of Seventh Powers et 1 (2011.7.18)]
[119. Equal Sums of Eighth Powers (2011.7.1)]
[118. Solutions of x^4 + y^4 + z^4 = w^n (2011.6.3)]
[117. EQUAL SUMS OF ELEVENTH POWERS (2011.5.6)]
[116. Solutions of x^4 + y^4 + z^4 = nw^3 (2011.4.1)]
[115. EQUAL SUMS OF SEVENTH POWERS (2011.3.4)]
[114. New solution: 3047 = 234115946^3 - 81713665^3 - 2* 183146068^3 (2011.2.16)]
[113. Generalized Taxicab Number Part 2 (2011.2.1)]
[112. Happy New Year! (2011.1.1)]
[111. Generalized Taxicab Number (2010.12.3)]
[110. x1^4 + x2^4 +...+ xn^4 = nw^2 (2010.11.1)]
[109. ON EQUAL SUMS OF NINTH POWERS Part 2 (2010.10.1)]
[108. Solutions of x^4 + y^4 + z^4 = nw^2 (2010.9.1)]
[107. Any rational number N is the sum and difference of four rational fourth powers
in an infinite number of ways. (2010.8.1)]
[106. ON EQUAL SUMS OF NINTH POWERS (2010.7.20)]
[105. Solutions of x^4 + y^4 = cz^2 (2010.8.6)(2010.8.5)(2010.7.1)]
[104. Solutions of X^3 + Y^3 + Z^3 = mW^3 (2010.6.1)]
[103. 3956= 3481598477^3 - 1323591649^3 - 2* 2711779804^3 (2010.5.11)]
8779= 64325766916^3 - 11325451823^3 - 2* 50962343435^3
[102. Solutions of x3 + y3 + z3 = tn (2010.5.1)]
[101. A4+ B4+ C4 = D4+ E4 Part 2 (2010.4.10)]
[91. Solutions of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 (2010.4.11)(2010.4.7)(2010.4.2)(2009.7.30)(2009.7.12)]
[100. x^4 + ay^4 = z^4 + bt^4 (2010.3.31)]
[99. 1 = (4x^4 + 1)^4 -(4x^4)^4 -(4x^3)^4 -(2x)^4 -(3x^2)^4 -(2x^2)^4 +(x^2)^4 (2010.3.26)]
[98. 2 = (8x^5-2x)^4 + (8x^4+1)^4 + (8x^4-1)^4 - (8x^5+2x)^4 - (4x^2)^4 (2010.3.10)]
[97. Ramanujan's identity (4x^5-5x)^4 + (6x^4-3)^4 + (4x^4+1)^4 - (4x^5+x)^4 - (2x^4-1)^4 = 81 (2010.3.3)]
[70. Solutions of a15+a25+a35 = b15+b25+b35 (2010.2.12) (2008.5.1)]
[96. Repunit (2010.1.13)]
[95. A Happy New Year! (2010.1.1)]
1. Cubic residue and x3+y3+z3=d(2005.4.2)
3. Ramanujan's cubic identity(2005.4.26)(2006.5.10)
6. A4+ B4+ C4= D4(Counterexample of Euler's conjecture) (2005.5.22)(2005.6.13)
8. A4+ B4+ C4+ D4+ E4= F4(Ramanujan's identities) (2005.6.19)
9. A4+ B4+ C4 = D4+ E4+ F4 (2005.6.25)
22. A4+ B4+ C4 = D4+ E4+ F4 PART 2 (2006.2.1)
23. A4+ B4+ C4+ D4 = E4+ F4 (2006.2.12)
24. A4+ B4+ C4 = D4+ E4 (2006.2.26)
25. A14+A24+A34+...+An4=B14+B24+B34+...+Bn4(May the Fourth with you!) (2006.3.19)
26. A5+ B5+ C5+ D5= E5+ F5+ G5 (2006.3.26)
27. A5+ B5+ C5+ D5= E5+ F5+ G5+ H5 (2006.3.26)
28. A15+A25+A35+...+An5=B15+B25+B35+...+Bn5 (2006.3.26)
29. Prime-generating polynomials(Fourth degree) (2006.3.31)
30. Representation by fourth powers (2006.5.2)
31. A13+A23+A33+...+An-13=An3(Ggeneralized Ramanujan's cubic identity) (2006.5.21)
32. 31116963 is expressed as a sum of three positive cubes by 20 ways (2006.5.27)
33. A4+ B4+ C4+ D4+ E4= F4(Ramanujan's identities, again) (2006.6.10)
34. A14+A24+A34+...+An-14=An4(Ggeneralized Ramanujan's 4th power identity) (2006.6.22)
35. a14+b14+c14 = a24+b24+c24 =..... = an4+bn4+cn4 (2006.6.22)
36. A15+ A25+ A35+ A45 = B15+ B25+ B35+ B45+ B55+ B65 (2006.8.4)
39. A15+ A25+ A35+ A45+ A55+ A65 = B15+ B25+ B35+ B45+ B55+ B65 (2006.9.23)
40. A15+ A25+ A35+ A45+ A55+ A65+ A75 = B15 (2006.10.14)
41. A15+ A25+ A35+ A45+ A55+ A65+ A75 = 2 (2006.10.27)
42. A17+ A27+ .... A87 = B17+ B27+ .... B87 (2006.11.10)
43. Solutions of 5.6.6, 6.8.8, 8.10.10, 9.10.10 and 10.12.12. (2006.11.18)
44. A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46 (2006.12.1)
45. A16+ A26+ A36+ A46 = B16+ B26+ B36+ B46 Part 2 (2006.12.14)
46. A Happy New Year! (2007.1.1)
47. A16+ A26+ ....+ A66 = B16+ B26+ ....+ B66 (2007.1.10)
48. A16+ A26+ ....+ A66 = B16+ B26+ ....+ B66 Part 2 (2007.1.21)
49. Current status of the solutions for A4=B4+C4+D4 (2007.1.26)
50. A property of the solution for a1p-1+a2p-1+....+anp-1=bp-1 (2007.2.4)
54. A16+ A26+ ....+ A46 = B16+ B26+ ....+ B46 (2007.3.28)
55. A16+ A26+ ....+ A46 = B16+ B26+ ....+ B46 Part 2 (2007.4.10)
56. A16+ A26+ ....+ A46 = B16+ B26+ ....+ B46 Part 3 (2007.5.1)
57. A16+ A26+ ....+ A76 = B16+ B26+ ....+ B76 (2007.5.20)
60. A17+ A27+ ....+ A57 = B17+ B27+ ....+ B57 (2007.6.15)
61. A18+ A28+ ....+ A78 = B18+ B28+ ....+ B78 (2007.7.16)
62. A18+ A28+ ....+ A68 = B18+ B28+ ....+ B68 (2007.8.5)
63. EQUAL SUMS OF SEVENTH POWERS (2007.8.20)(2007.10.15)
64. A16+ A26+ ....+ A56 = B16+ B26+ ....+ B56 (2007.10.15)
65. A16+ A26+ ....+ A66 = B16+ B26+ ....+ B66 (2007.10.15)
66. 58219814004+153558313604+1409765514=154345478014 (2007.10.24)
67. A Happy New Year! (2008.1.1)
68. Sums of three cubes (2008.2.7)(2008.1.15)(2008.1.8)(2008.1.5)
69. Sierpinski problem (2008.4.11)
70. Solutions of a15+a25+a35 = b15+b25+b35 (2010.2.12) (2008.5.1)
71. 4 + 27 = 31!? (2008.5.12)
72. 49875884196554+24804526756004+5020388539764=50622976992574(2008.5.15)
73. N=5,6,and 7 mod 8 is always a congruent number (2008.6.1)
74. Solve the Diophantine equation like Diophantus (2008.6.15)
75. New solutions of n=x3+y3+2z3 (2008.7.31)(2008.7.13)(2008.6.22)
76. 478867402721149764 + 88134256704402404 + 568278133081117854 = 629405169034106014 (2008.8.13)]
35790871473754404 +148900264334684714 + 185659451142167204 = 202495067095797214 (2008.8.13)]
77. 12597684734=11667058404+8593964554+5889033364 (2008.9.1)]
8738221214=7693212804+6067108714+5584244404
17878823374=16629976634+12377969604+6863980004
18717138574=15935130804+15535564404+926224014
78. Table of the solutions for A4+B4+C4=D4 (2008.9.1)]
79. 1300643009914004+ 4408049425801604+ 5148181012992894= 5736463218719614 (2008.9.15)]
80. Table of the solutions for A4+B4+C4=D4 Part 2 (2008.10.5)]
81. Every integer is a sum of four cubes of integers? (2008.11.1)]
82. Table of the solutions for x3+by3+cz3=0. (2008.12.3)]
83. A Happy New Year! (2009.1.1)]
84. A4=B4+C4+D4+E4 (2009.2.10)]
85. x4+y4+z4 = nw4 (2009.3.1)]
86. A4+ B4+ C4+ D4+ E4= F4 (Ramanujan's identities, again x 2) (2009.3.16)]
87. Representation by fourth powers (2009.5.1)]
88. 9033964577482532388059482429398457291004947925005743028147465732645880^4 +
4417264698994538496943597489754952845854672497179047898864124209346920^4 +
1439965710648954492268506771833175267850201426615300442218292336336633^4 =
9161781830035436847832452398267266038227002962257243662070370888722169^4 (2009.6.1)]
89. Representation by fourth powers Part 2 (2009.6.15)]
90. Solutions of a16+a26+a36 = b16+b26+b36 (2008.6.26)
91. Solutions of a^4 + b^4 + c^4 + d^4 = (a+b+c+d)^4 (2009.7.30)(2008.7.12)
92. Smallest number that is a sum of two n-th powers of positive
rationals but not of two n-th powers of positive integers. (2009.9.6)
93. Solutions of X4 + Y4 = 4481Z2. (2009.11.1)
94. On factors of Mersenne numbers. (2009.12.10)
95. A Happy New Year! (2010.1.1)
96. Repunit (2010.1.13)
97. Ramanujan's identity (4x^5-5x)^4 + (6x^4-3)^4 + (4x^4+1)^4 - (4x^5+x)^4 - (2x^4-1)^4 = 81 (2010.3.3)
98. 2 = (8x^5-2x)^4 + (8x^4+1)^4 + (8x^4-1)^4 - (8x^5+2x)^4 - (4x^2)^4 (2010.3.10)
99. 1 = (4x^4 + 1)^4 -(4x^4)^4 -(4x^3)^4 -(2x)^4 -(3x^2)^4 -(2x^2)^4 +(x^2)^4 (2010.3.26)
100. x^4 + ay^4 = z^4 + bt^4 (2010.3.31)
101. A4+ B4+ C4 = D4+ E4 Part 2 (2010.4.10)
102. Solutions of x3 + y3 + z3 = tn (2010.5.1)
103. 3956= 3481598477^3 - 1323591649^3 - 2* 2711779804^3 (2010.5.11)
8779= 64325766916^3 - 11325451823^3 - 2* 50962343435^3
104. Solutions of X^3 + Y^3 + Z^3 = mW^3 (2010.6.1)
105. Solutions of x^4 + y^4 = cz^2 (2010.7.1)
106. ON EQUAL SUMS OF NINTH POWERS (2010.7.20)
107. Any rational number N is the sum and difference of four rational fourth powers
in an infinite number of ways. (2010.8.1)
108. Solutions of x^4 + y^4 + z^4 = nw^2 (2010.9.1)
109. ON EQUAL SUMS OF NINTH POWERS Part 2 (2010.10.1)
110. x1^4 + x2^4 +...+ xn^4 = nw^2 (2010.11.1)
111. Generalized Taxicab Number (2010.12.3)
112. Happy New Year! (2011.1.1)
113. Generalized Taxicab Number Part 2 (2011.2.1)
114. New solution: 3047 = 234115946^3 - 81713665^3 - 2* 183146068^3 (2011.2.16)
115. EQUAL SUMS OF SEVENTH POWERS (2011.3.4)
116. Solutions of x^4 + y^4 + z^4 = nw^3 (2011.4.1)
117. EQUAL SUMS OF ELEVENTH POWERS (2011.5.6)
118. Solutions of x^4 + y^4 + z^4 = w^n (2011.6.3)
119. Equal Sums of Eighth Powers (2011.7.1)
120. Equal Sums of Seventh Powers et 1 (2011.7.18)
121. Equal Sums of Sixth Powers et 1 (2011.8.1)
122. Equal Sums of Fifth Powers et 1 (2011.8.23)
123. Equal Sums of Fourth Powers et 1 (2011.9.13)
124. Equal Sums of Eighth Powers et 1 (2011.10.11)
125. New parametric solutions of x^3 + y^3 + z^3 = t^2 (2011.11.16)
126. Any non-zero rational number m is the sum of three rational cubes
in an infinite number of non-trivial ways. (2011.12.6)
127. Happy New Year! (2012.1.3)
128. Equal sums and equal products (2012.2.1)
129. Equal Sums of Fifth Powers (2012.3.2)
130. A collection of the parametric solutions for equal sums of fifth powers (2012.4.2)
131. A collection of the parametric solutions for equal sums of sixth powers (2012.5.3)
132. Equal Sums of Eighth Powers Part 3 (2012.6.11)
133. Solutions of x^4 + y^4 + z^4 = w^2 (2012.7.2)
134. New solution: 4388 = -18105946705^3 + 18108094211^3 - 2*1018372219^3 (2012.7.12)
135. Equal Sums of Eighth Powers Part 4 (2012.8.10)
136. Solutions of ax^6 + by^3 = cz^2 (2012.9.10)
137. Many solutions of n=x^3 + y^3 + 2z^3 (2012.10.18)(2012.9.24)
138. Equal Sums of Seventh Powers (2012.11.30)
139. Happy New Year! (2013.1.1)
140. A parametric solution of X^k + Y^k + Z^k + W^k + S^k = {U^2,V^4} for k=2,4 (2013.2.1)
141. Solutions of x^2 + y^4 = 5z^4 (2013.3.1)
142. Solutions of x^4 + ay^4 = z^4 + bt^4 (2013.4.1)
143. Solutions of x^4 + ax^2y^2 + y^4 = u^4 + au^2v^2 + v^4 (2013.5.1)
144. Four positive numbers such that the sum of any two is a cube (2013.5.15)
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