Suppose the smaller of the two even natural numbers is 2x. Then the other one is 2x + 2
The sum of their squares is (2x)2 + (2x + 2)2 = 4x2 + 4x2 + 8x + 4 = 8x2 + 8x + 4
You want 8x2 + 8x + 4 < 1000
that is 8x2 + 8x - 996 < 0
or 2x2 + 2x - 249 < 0
x < {-2 + sqrt[22 - 4*2*(-249)]}/(2*2)
= {-2 + sqrt[4 + 1992]}/(4)
= {-2 + sqrt[1996]}/(4)
= {-2 + 44.68}/4
= 42.68/4
= 21.34
So x = 20
Then, the sum is 202 + 222 = 400 + 484 = 884
202 + 222 = 400 + 484 = 884
There are no three consecutive numbers that are squares. Otherwise, there are an infinite sets of squares of three consecutive numbers: for example, {1,4,9}, or {4,9,16} or {576, 625, 676}
5
12 and 13
The numbers are 12 and 14.
202 + 222 = 400 + 484 = 884
There are no three consecutive numbers that are squares. Otherwise, there are an infinite sets of squares of three consecutive numbers: for example, {1,4,9}, or {4,9,16} or {576, 625, 676}
16,25,36 add to 77. They are the squares of consecutive numbers 4,5,6
Natural numbers which are the scales of some natural numbers are perfect squares
The numbers are 13 and 14.
5
8
5
15 and 16
17 and 18
62 and 63
12 and 13