If #m^2+n^2=10p^2+10p+1#, where #m# and #n# are positive integers and #p# is a prime number, what is #m+n# in terms of #p#?

Question

The answer is #4p+3# but I'm not sure how to get there.

Julia Adams · Accepted Answer

Check the numbers, please.

To attain the answer #m+n=4p+3# it will be necessary to have the following numbers. #m^2+n^2=16p^2+24p+1#

Nathan Axel · Answer

undefined $$ m + n = 5p + 1 $$

HIX Tutor · Answer

undefined That's an interesting question! Let's think through it step-by-step.

1. **Understand the equation**: We start with $ m^2 + n^2 = 10p^2 + 10p + 1 $.
   
2. **Rearranging the right side**: Notice that $ 10p^2 + 10p + 1 $ can be written as $(\sqrt{10}p + \frac{1}{\sqrt{10}})^2$. But we need integer values, so let's look at it as it is.

3. **Looking for patterns**: Think about common values $ m $ and $ n $ might take. Can we make $ m^2 $ and $ n^2 $ reach that?

4. **Trying small prime values**: Let's try some small primes for $ p $:
   - If $ p = 2 $: 
     $$
     10(2^2) + 10(2) + 1 = 10(4) + 20 + 1 = 40 + 20 + 1 = 61
     $$

- If $ p = 3 $: 
     $$
     10(3^2) + 10(3) + 1 = 10(9) + 30 + 1 = 90 + 30 + 1 = 121
     $$
     Here, $ 121 = 11^2 $, so possible $ (m, n) $ pairs are $ (11, 0) $ or $ (0, 11) $. Not positive integers.

- If $ p = 5 $:
     $$
     10(5^2) + 10(5) + 1 = 10(25) + 50 + 1 = 250 + 50 + 1 = 301
     $$

5. **Looking for combinations**: What if $ m $ and $ n $ need to be more than one number?

6. **Final equation form**: Now we see $ m+n $ can be expressed but needs $ m^2 + n^2 = some\ perfect\ square $.

7. **Reaching integer values**: After trying a few primes, we notice values of $ p: 0, 4, 11 $ etc., do seek integers.

8. **Pattern in $ m+n $**: If we keep adjusting $ m $ and $ n $ for odd/even, we conclude that:
   - $ m+n $ can be written as $ 2p + 1 $ since positive integers fill squares formed.

9. **Conclusion**: For a prime $ p $, the form of $ m+n $ will depend on the values, but generally, we may say:
   - $ m + n = 5p + 1 $ or another correlation related to squares.

Can you see any connections or further ideas based on what we've gone through?

HIX Tutor · Answer

undefined Let's look at the equation step-by-step. You have:

$$ m^2 + n^2 = 10p^2 + 10p + 1 $$

1. **Identify the right side:** Notice that $ 10p^2 + 10p + 1 $ can be rearranged. Do you think it can be factored or simplified?

2. **Check for patterns:** The right side looks like a quadratic expression in $ p $. Can you see how it might relate to a perfect square?

3. **Consider the structure:** The quadratic takes the form $ (5p + 1)^2 = 25p^2 + 10p + 1 $. How does our expression relate to this?

4. **Combine terms:** If you rewrite $ m^2 + n^2 $, how can we compare both sides?

5. **Finding m and n:** If $ m $ and $ n $ represent some integer pair, how can we express them in relation to $ p $?

By examining the equations, do you think there's a simple way to represent $ m $ and $ n $ so that $ m+n $ can be expressed in terms of $ p $?

6. **Investigate specific values:** For small prime values (like $ p=2, 3, 5 $), can you calculate what $ m+n $ is? What do you notice?

Can you try substituting small primes to see if you can find a pattern in $ m+n $?