Which is the smallest number that can be written in two ways as the sum oftwo perfect squares

To find the smallest number that can be expressed as the sum of two perfect squares in two different ways, we start by recalling a few mathematical concepts regarding sums of squares.

Step 1: Understanding Perfect Squares

A perfect square is an integer that is the square of another integer. The first few perfect squares are:

• $0^2 = 0$
• $1^2 = 1$
• $2^2 = 4$
• $3^2 = 9$
• $4^2 = 16$
• $5^2 = 25$
• $6^2 = 36$, and so forth.

Step 2: Finding Sums of Two Perfect Squares

We seek integers $a$, $b$, $c$, and $d$ such that: $n = a^2 + b^2 = c^2 + d^2$ where $(a, b)$ and $(c, d)$ are different pairs.

Step 3: Exploring Combinations

We can systematically calculate the sums of two perfect squares:

1. $1^2 + 0^2 = 1$
2. $1^2 + 1^2 = 2$
3. $2^2 + 0^2 = 4$
4. $2^2 + 1^2 = 5$
5. $2^2 + 2^2 = 8$
6. $3^2 + 0^2 = 9$
7. $3^2 + 1^2 = 10$
8. $3^2 + 2^2 = 13$
9. $3^2 + 3^2 = 18$
10. $4^2 + 0^2 = 16$
11. $4^2 + 1^2 = 17$
12. $4^2 + 2^2 = 20$
13. $4^2 + 3^2 = 25$
14. $4^2 + 4^2 = 32$
15. $5^2 + 0^2 = 25$
16. $5^2 + 1^2 = 26$
17. $5^2 + 2^2 = 29$
18. $5^2 + 3^2 = 34$
19. $5^2 + 4^2 = 41$
20. $5^2 + 5^2 = 50$

Step 4: Identifying the Smallest Number

We seek pairs:

• From the sums we calculated, we find:
  • $1^2 + 2^2 = 1 + 4 = 5$
  • $2^2 + 1^2 = 4 + 1 = 5$

Continuing to check:

• $3^2 + 4^2 = 9 + 16 = 25$
• $0^2 + 5^2 = 0 + 25 = 25$

Eventually:

• The smallest number found:
  • $5$ can be expressed as
    • $1^2 + 2^2 = 5$
    • $2^2 + 1^2 = 5$
• The next smallest number is $25$.

Final Answer

Thus, the smallest number that can be written in two different ways as the sum of two perfect squares is $5$ and follows $25$ as per the unique representations.