I bet u can't answer this?

Question

My question-

Find 4 no.s such that sum of any 2 is a perfect square and sum of all 4 is also a perfect square.

all the no.s are different....

Answer 1

I bet that I can.

The set {386, 2114, 3970, 10430} works:

386 + 2114 = 2500 = 50^2
386 + 3970 = 4356 = 66^2
386 + 10430 = 10816 = 104^2
2114 + 3970 = 6084 = 78^2
2114 + 10430 = 12544 = 112^2
3970 + 10430 = 14400 = 120^2
386 + 2114 + 3970 + 10430 = 16900 = 130^2

Other sets that work include:
{590, 4594, 5810, 17906}
{1544, 8456, 15880, 41720}
{2345, 5936, 35680, 61664}

There are probably many other sets of four which work (perhaps infinitely many).

Answer 2

0
a+b, c+d
a+c, b+d
a+d, b+c

it seems to me we only need to find a largest square that have 3 ways of being sum of 2 other squares.

i tried to have the smallest solution, but i don't know if this really would be the smallest. but you can always pick any 3 pytagorean triple(?) to do this.
3, 4, 5
5, 12, 13
7, 24, 25

find LCM or whatever you call that, and syncronize them (hypotenuses) together.
195, 260, 325
125, 300, 325
91, 312, 325

a+b = 91^2 = 8281 ,
c+d = 312^2 = 97344

a+c = 125^2 = 15625 ,
b+d = 300^2 = 90000

a+d = 195^2 = 38025 ,
b+c = 260^2 = 67600

2d =(c+d) + (b+d) - (b+c)
= 97344 + 90000 - 67600
= 119744

d = 59872
c = 97344 - 59872 = 37472
b = 90000 - 59872 = 30128
a = 38025 - 59872 = -21847

well, one of my answers turns out to be a negative. but since you said nothing about that in your question on polarity, so i'll just let that be.


edit

and the mathematician's triples are
5, 12, 13
3, 4, 5 and to my surprise,
33, 56, 65

LCM(?) should give you 65 but as it happens, 65 also gives 1 negative smallest.

Answer 3

As the first answerer I came up with beatiful solutions like:
1, 8, 8, 8
4, 32, 32, 32
9, 72, 72, 72
18, 18, 18, 46

But since you said the numbers ought to be different, then the solution is quite complicated, let me work it out more...

CAN ANY OF THE NUMBERS BE NEGATIVE?

Answer 4

zero, zero, zero, zero

UPDATE:

Darn, the Mathematician beat me to it by half an hour.
But I can't generate them, I just coded it up and tried all combinations of 4 numbers up to some maximum. (I used some bit manipulations, and figured I could complete to 62,500 by morning.)

This is unsatisfying.
for (a = 0; a <= MAX-3; a++)
{
for (b = a+1; b <= MAX-2; b++)
{
if (!is_sq(a+b))
continue;
for (c = b+1; c <= MAX-1; c++)
{
if ((!is_sq(a+c))
|| (!is_sq(b+c)))
continue;
for (d = c+1; d <= MAX; d++)
{
if ((is_sq(a+d))
&& (is_sq(b+d))
&& (is_sq(c+d))
&& (is_sq(a+b+c+d)))
{
printf("%d, %d, %d, %d:\n", a, b, c, d);
printf(" %d, %d, %d, %d, %d, %d, and %d are squares\n",
a+b,a+c,a+d,b+c,b+d,c+d,a+b+c+d);
}
}
}
}
}

Answer 5

i think you are a perfect square