Only Perfect squares

Number Theory Level 5

$a,b,c,$ and $d$ are distinct positive integers which satisfy the following properties:

the sum of any two of them is a perfect square, and
the sum of the four numbers is also a perfect square.

Find the smallest possible value of $\sqrt{a+b+c+d}$ .

The answer is 130.

1 solution

Ossama Ismail
Feb 20, 2017

The smallest possible values that satisfy the given conditions are:-

$a=386 \\ b=2114 \\ c=3970 \\ d=10430$

$a+b = 2500 = 50^2 \\ a+c =4356 = 66^2 \\ a+d = 10816 = 104^2 \\ b+c = 6084 = 78^2 \\b+d=12544=112^2\\ c+d = 14400 = 120^2 \\ S =a+b+c+d =16900 =130^2 \\ \sqrt{S} = 130$

$a=386 \\ b=2114 \\ c=3970 \\ d=10430$

How did you find these numbers?

Pi Han Goh - 4 years, 3 months ago

When (a+b+c+d) is a perfect square, the answer can be found by using Pythagorean and the number has prime factors, 2 or in the form of n%4.=1

130=2 x 5 x 13 is one of these numbers. will get back to you later.

Ossama Ismail - 4 years, 3 months ago

Nice, that's the idea. We want at least 3 (unordered) pairs of positive integer solutions to $x^2 + y^2 = s^2$ , and we can use the characterization to tell that $s = 2 \times 5 \times 13$ is the minimum possible value of $s$ .

We then have 4 (linearly independent) equations in 4 unknowns, which lead to a unique solution (that we has to verify has positive integer solutions).

Calvin Lin Staff - 4 years, 3 months ago

@Calvin Lin – What do you think about adding the word 'distinct' to the question? It would otherwise appear as though (1,8,8,8) satisfies the conditions.

Abel McElroy - 4 years, 3 months ago

@Abel McElroy – distinct was added to the question.

Ossama Ismail - 4 years, 3 months ago

@Ossama Ismail – Nice! thanks. great question! thanks for posting it.

Abel McElroy - 4 years, 3 months ago

@Ossama Ismail - I added the $b+d=12544=112^2$ line. Hope you don't mind. I think the next largest set of 4 distinct Natural Numbers is {872, 2377, 9944, 21032} which would give rise to the answer $\boxed{185}$

Woops! The second set of 4 distinct Natural Numbers is {590, 4594, 5810, 17906} which would give rise to the answer $\boxed{170}$

Bob Kadylo - 4 years, 1 month ago

Only Perfect squares

The answer is 130.

1 solution

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