To find integers such that

with one more term on the left than on the right

Note the identity:

we have

To find integers such that

with one more term on the left than on the right

Note the identity:

we have

separated into 3 sets:

the sum in any set is the same

sum of squares of the numbers is the same

Take the numbers from **1** to **64**, separate them into **4** sets such that the sum and sum of squares (or cubes) of the numbers in any set are the same as for the other sets.

N.B. The generalization is a bit difficult.

the pattern continues.

Can you explain why?

So

Factorize

Find integers that have 2 representations as a sum of powers

where and are positive integers.

and

E.g.

Can you find more examples?

**Part #1**

To find integers such that

Let’s take six integers

and

if

E.g.

x + y = 2 + 3, b = 1, c = 6, x = 2, y = 3, b + c = 7

and

——————————————

**Part #2**

From an integer N with 2 representations as a sum of 3 squares to an integer M with 2 representations as a sum of 4 squares

If

then, consider the two sets of integers:

and

It’s clear that

——————————————

**Part #3**

So

If we have consecutive integers:

, , ,

And,

If , , ,

To find integers such that

with one more term on the left than on the right

Note the identity:

we have