A, B, and C are three positive integers in arithmetic sequence with A < B < C.
Do there exist three distinct positive integers P, Q, and R that satisfy this system of equations:
- A+B*C = P^2
- B+C*A = Q^2
- C+A*B = R^2?
Provide valid reasoning for your answer.
Putting the equations into standard arithmetic progression format, noting the symmetry of the system and observing that the common difference d must be smaller than a for A,B,C all to be positive, then we can select a pair of equations such that:
a^2+ad+a-d = say q^2 and a^2-ad+a+d = say r^2, with positive difference 2d(a-1).
Now a^2+ad+a-d = q^2 (a bit bigger than a^2) and a^2-ad+a+d = r^2, (a bit smaller than a^2):
(a-1)d = a(2n-1)+n^2 for q^2 (to make it a perfect square for some integer n)
(1-a)d = a(-2m-1)+m^2 for r^2 (to make it a perfect square for some integer m<n)
But (1-a)d =-(a-1)d, so a(2n-1)+n^2 = a(2m+1)-m^2, or m^2+n^2 = 2a(m-n+1)
Here LHS is positive, but RHS cannot exceed zero, so there is a contradiction.
Hence no such distinct positive integers P,Q,R exist.
Edited on August 10, 2023, 1:39 am
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Posted by broll
on 2023-08-10 01:33:43 |
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