Math Help - Sum of 2 squares

1. Sum of 2 squares

Sorry before I tell you the question here was the question that led to this question.

Show that (m^2+1)(n^2+1)=(m+n)^2+(mn-1)^2

Done that successfully.

Using this results, write 500050 as the sum of 2 square numbers.

I have no idea how to make a knowledgeable guess at the two squares so I've tried using random (well, not exactly, generic random numbers) to get an answer but to no avail

2. Originally Posted by Mukilab
Sorry before I tell you the question here was the question that led to this question.

Show that (m^2+1)(n^2+1)=(m+n)^2+(mn-1)^2

Done that successfully.

Using this results, write 500050 as the sum of 2 square numbers.

I have no idea how to make a knowledgeable guess at the two squares so I've tried using random (well, not exactly, generic random numbers) to get an answer but to no avail
Try m=7.

50 is very close to 49 and stands out that way..

3. Hello, Mukilab!

Show that: . $(m^2+1)(n^2+1)\:=\m+n)^2+(mn-1)^2" alt=" (m^2+1)(n^2+1)\:=\m+n)^2+(mn-1)^2" />

Done that successfully. . Good!
Using this results, write 500,050 as the sum of 2 square numbers.

Note that: . $500,050 \;=\;(50)(10,\!001) \;=\;(7^2+1)(100^2+1)$

$\text{Let }m = 7,\;n = 100 \text{ in the formula:}$

. . $(7^2 + 1)(100^2 + 1) \;=\;(7 + 100)^2 + (7\cdot100 - 1)^2 \;=\;107^2 + 699^2$

4. m=7 is a really nice guess
OK. $500050=10001*50$
and $10001=10000+1$
and $50=49+1$
Apply in your equation: $(m^2+1)(n^2+1)=(m+n)^2+(mn-1)^2$
Then....

5. That would make n a decial (10sqrt10)

6. Originally Posted by Soroban
Hello, Mukilab!

Note that: . $500,050 \;=\;(50)(10,\!001) \;=\;(7^2+1)(100^2+1)$

$\text{Let }m = 7,\;n = 100 \text{ in the formula:}$

. . $(7^2 + 1)(100^2 + 1) \;=\;(7 + 100)^2 + (7\cdot100 - 1)^2 \;=\;107^2 + 699^2$

Thanks Soroban!

Sorry, only saw the first post at the start

Any tips for future questions such as this?? Take away the integers so I'm left with the algebraic numbers (as you did with 49 and 100)?

Thanks again

7. Originally Posted by Mukilab
That would make n a decial (10sqrt10)
Maybe you entered 50050 / 50 instead of 500050 / 50? m=7 gives n=100 as shown above.

8. Originally Posted by Mukilab
Thanks Soroban!

Sorry, only saw the first post at the start

Any tips for future questions such as this?? Take away the integers so I'm left with the algebraic numbers (as you did with 49 and 100)?

Thanks again
Another way to go about expressing integers as sums of two squares is knowing that: the set of integers expressible as the sum of two squares is closed under multiplication, and an odd prime is expressible as the sum of two squares if and only if it is congruent to 1 (mod 4). See here and here.