Does there exist any positive integral solution of the equation given below?
16(X³+Y³) + 12(X²+Y²) + 3(X+Y) = 2,138,156,388
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Submitted by K Sengupta
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Rating: 3.0000 (2 votes)
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Solution:
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At the outset, it may be pertinent to mention that amongst all the responses received till date, only goFish has conclusively demonstrated the non-existance of any positive integral solution to the problem, by means of a brilliant synergy of number theoretic methods and analytical reasoning.
SOLUTION TO THE PROBLEM:
No positive integral solution exists for the equation under reference.
EXPLANATION :
Multiplying both sides of the given equation by 4 and adding 2 , we obtain:
P^3 + Q^3 = 8,552,625,554 where P = 4X + 1 and Q = 4Y + 1.
Now, we know that the cube of any positive integer when divided by 7 leaves a remainder equivalent to any one of 0,1 and 6. Hence, the sum of cube of two positive integers when divided by 7 leaves a remainder equivalent to any one of 0,1,2,5 and 6 and consequently integers having remainders 3 or 4 upon division by 7 are not expressible as the sum of cube of two positive integers.
Now, 8,552,625,554 leaves a remainder of 4 upon division by 7.
Accordingly, it follows that the said integer (8,552,625,554 ) cannot be expressed as the sum of cube of two positive integers. This contradicts the provisions of the problem under reference.
Consequently, no positive integral solution exists for the equation under reference.
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