Computational Number Theory

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Here are computational Number Theory topics

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Update 2014.4.5

New!

           [154. Any rational number N is the sum and difference of four rational fourth powers
               in an infinite number of ways Part 2  (2014.4.5)]
           [153. Equal Sums of Eighth Powers  Part 5(2014.3.3)]
           [152. Solutions  of  x^3 + y^3 + z^3 = t^n  Part 2(2014.1.31)]
           [151. Happy New Year!  (2014.1.1)]
           [150. Solutions of sextic diophantine equation(2013.11.1)]
           [149. Solutions of x^3 + dy^3 + d^2z^3 - 3dxyz = 1(2013.9.28)]
           [148. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 2(2013.9.12)(2013.8.17)]
           [147. Solutions of A^4 + hB^4 = C^4 + hD^4  (2013.8.1)]
           [146. Solution of X^k + Y^k = {U^2,V^4} for k=1,2  (2013.7.12)]
           [145. Three positive numbers such that the sum and difference of any two are squares  (2013.6.14)]
           [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)]
          


Topics

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)

145. Three positive numbers such that the sum and difference of any two are squares (2013.6.14)

146. Solution of X^k + Y^k = {U^2,V^4} for k=1,2 (2013.7.12)

147. Solutions of A^4 + hB^4 = C^4 + hD^4 (2013.8.1)

148. Solutions of A^4 + hB^4 = C^4 + hD^4 Part 2 (2013.8.17)

149. Solutions of x^3 + dy^3 + d^2z^3 - 3dxyz = 1 (2013.9.28)

150. Solutions of sextic diophantine equation (2013.11.1)

151. Happy New Year! (2014.1.1)

152. Solutions of x^3 + y^3 + z^3 = t^n Part 2 (2014.1.31)

153. Equal Sums of Eighth Powers Part 5 (2014.3.3)

154. Any rational number N is the sum and difference of four rational fourth powers in an infinite number of ways Part 2 (2014.4.5)



















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