+0

# Number Theory

+1
16
1
+81

How many positive integers less than 2024 cannot be expressed as the difference of the squares of two positive integers?

What I have deduced so far is that the number cannot be a perfect square, it cannot be an odd number, and it has to be in the form 2 * (some other numbers). The first two has to be only to the power of 1, if it is to the power of two or higher, than it can also be written as two squares. Could anyone help me procced from here? Thanks!

Jan 12, 2024

#1
+129799
+1

b^2  - a^2     b^2   a^2

1                 1      0       (violates  our rules.... a,b must be positive)

2                    NP

3                 2      1

4                 2      0      (violates our  rules)

5                 3      2

6                   NP

7                4       3

8                3       1

9                5       4

10                 NP

11               6       5

12               4       2

13               7       6

14                NP

15                8      7

16                5      3

17                9      8

18                NP

It appears that all odds (except 1) can  be made

And it appears that  all  evens can  be made except  4 and  the ones of the form 2 + 4(n -1)   where n =  1,2,3,4....

You should check  for yourself  with a few values !!!

Note that  when n = 506     2 + 4 (506 - 1) =  2022

So.....the answer  should  be     2 values = (1,4) + 506 values =   508 integers cannot be made

Hope that helps....

Jan 12, 2024
edited by CPhill  Jan 12, 2024
edited by CPhill  Jan 12, 2024