1729 again

Number Theory Level 4

Find the number of ordered pairs of positive integers ( a , b ) (a,b) such that

  • a 2 + b 2 = c a^2 + b^2 = c
  • a 3 + b 3 = f ( a + b ) a^3 + b^3 = f ( a + b)
  • c − f = 1729 c - f = 1729 .


The answer is 8.

Shivam Jadhav by Shivam Jadhav , 22, India

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1 solution

Department 8
Sep 10, 2015

a 2 + b 2 = c a 3 + b 3 = f ( a + b ) a^{2}+ b^{2}=c \\ a^{3}+b^{3}=f(a+b)

c − f = 1729 a 2 + b 2 − a 3 + b 3 a + b = 1729 a 2 + b 2 − ( a 2 + b 2 − a b ) = 1729 a b = 1729 c-f=1729 \\ a^{2}+b^{2}-\frac{a^{3}+b^{3}}{a+b}=1729 \\ a^{2}+b^{2}-(a^{2}+b^{2}-ab)=1729 \\ ab=1729

1729 has 4 positive factors and 8 positive and negative factors. Answer is 8 \boxed{8}

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Actually, 1729 has 8 positive factors itself. i.e. 1,7,13,19,91,133,247,1729.

Mehul Arora - 5 years, 9 months ago

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