Determine the pair (a,b) of positive integers such that each of a/b, a/b+5, and a/b-5 is the square of a rational number.
The (a,b) pair that produces the fractions reduced to lowest terms is (1681, 144).
1681/144 = (41/12)^2
(1681 + 5*144)/144 = 2401/144 = (49/12)^2
(1681 - 5*144)/144 = 961/144 = (31/12)^2
Square multiples of these work also.
clearvars
sqs=[1:700].^2;
for asub=1:700
a=sym(sqs(asub));
for bsub=1:700
b=sym(sqs(bsub));
if a/b>=5
t1=a/b+5; t2=a/b-5;
[n,d]=numden(t1);
if round(sqrt(n))^2==n
if round(sqrt(d))^2==d
[n,d]=numden(t2);
if round(sqrt(n))^2==n
if round(sqrt(d))^2==d
disp([a,b, sqrt(a),sqrt(b), a/b])
end
end
end
end
end
end
end
squaare
roots
a b a b reduced
[1681, 144, 41, 12, 1681/144]
[6724, 576, 82, 24, 1681/144]
[15129, 1296, 123, 36, 1681/144]
[26896, 2304, 164, 48, 1681/144]
[42025, 3600, 205, 60, 1681/144]
[60516, 5184, 246, 72, 1681/144]
[82369, 7056, 287, 84, 1681/144]
[107584, 9216, 328, 96, 1681/144]
[136161, 11664, 369, 108, 1681/144]
[168100, 14400, 410, 120, 1681/144]
[203401, 17424, 451, 132, 1681/144]
[242064, 20736, 492, 144, 1681/144]
[284089, 24336, 533, 156, 1681/144]
[329476, 28224, 574, 168, 1681/144]
[378225, 32400, 615, 180, 1681/144]
[430336, 36864, 656, 192, 1681/144]
[485809, 41616, 697, 204, 1681/144]
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Posted by Charlie
on 2025-04-04 08:59:26 |
computer solution
