Question 1209456: Odd Perfect Numbers.
Descartes suggested (and later confirmed by Euler) that if an odd perfect number does exist, it must be of the form 𝑁=𝑃𝑀^2 where 𝑃 is prime and M a positive
odd >1.
Because a perfect number equals the sum of it's proper devisers, we can rewrite the above equation as:
𝑁=𝑃𝑀^2=𝑃+𝑃𝑀+𝑀^2+𝑀+1
𝑃*𝑀^2-𝑃𝑀-𝑃=𝑀^2+𝑀+1
𝑃(𝑀^2-𝑀-1) = 𝑀^2+𝑀+1
𝑃=(𝑀^2+𝑀+1)/(𝑀^2−𝑀−1)
It's clear from the last derivation, as M grows, in the limit P will tend to 1,
because the values of M+1 and M-1 would be negligible compared to M^2 as M grows. This mean 𝑁=𝑃𝑀^2 has no positive integer solutions.
Was Descartes wrong or am I missing something?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52778)   (Show Source):
You can put this solution on YOUR website! .
With all my respect to your writing, I copied-pasted it into the Google search engine.
The engine created a solution produced by AI (the (or an) artificial intelligence).
You may read this AI solution under this link
https://www.google.com/search?q=Odd+Perfect+Numbers.+Descartes+suggested+(and+later+confirmed+by+Euler)+that+if+an+odd+perfect+number+does+exist%2C+it+must+be+of+the+form+%F0%9D%91%81%3D%F0%9D%91%83%F0%9D%91%80%5E2+where+%F0%9D%91%83+is+prime+and+M+a+positive+odd+%3E1.+Because+a+perfect+number+equals+the+sum+of+it%27s+proper+devisers%2C+we+can+rewrite+the+above+equation+as%3A+%F0%9D%91%81%3D%F0%9D%91%83%F0%9D%91%80%5E2%3D%F0%9D%91%83%2B%F0%9D%91%83%F0%9D%91%80%2B%F0%9D%91%80%5E2%2B%F0%9D%91%80%2B1+%F0%9D%91%83*%F0%9D%91%80%5E2-%F0%9D%91%83%F0%9D%91%80-%F0%9D%91%83%3D%F0%9D%91%80%5E2%2B%F0%9D%91%80%2B1+%F0%9D%91%83(%F0%9D%91%80%5E2-%F0%9D%91%80-1)+%3D+%F0%9D%91%80%5E2%2B%F0%9D%91%80%2B1+%F0%9D%91%83%3D(%F0%9D%91%80%5E2%2B%F0%9D%91%80%2B1)%2F(%F0%9D%91%80%5E2%E2%88%92%F0%9D%91%80%E2%88%921)+It%27s+clear+from+the+last+derivation%2C+as+M+grows%2C+in+the+limit+P+will+tend+to+1%2C+because+the+values+of+M%2B1+and+M-1+would+be+negligible+compared+to+M%5E2+as+M+grows.+This+mean+%F0%9D%91%81%3D%F0%9D%91%83%F0%9D%91%80%5E2+has+no+positive+integer+solutions.+Was+Descartes+wrong+or+am+I+missing+something%3F&rlz=1C1CHBF_enUS1071US1071&oq=Odd+Perfect+Numbers.+Descartes+suggested+(and+later+confirmed+by+Euler)+that+if+an+odd+perfect+number+does+exist%2C+it+must+be+of+the+form+%F0%9D%91%81%3D%F0%9D%91%83%F0%9D%91%80%5E2+where+%F0%9D%91%83+is+prime+and+M+a+positive++odd+%3E1.+Because+a+perfect+number+equals+the+sum+of+it%27s+proper+devisers%2C+we+can+rewrite+the+above+equation+as%3A++%F0%9D%91%81%3D%F0%9D%91%83%F0%9D%91%80%5E2%3D%F0%9D%91%83%2B%F0%9D%91%83%F0%9D%91%80%2B%F0%9D%91%80%5E2%2B%F0%9D%91%80%2B1+%F0%9D%91%83*%F0%9D%91%80%5E2-%F0%9D%91%83%F0%9D%91%80-%F0%9D%91%83%3D%F0%9D%91%80%5E2%2B%F0%9D%91%80%2B1+%F0%9D%91%83(%F0%9D%91%80%5E2-%F0%9D%91%80-1)+%3D+%F0%9D%91%80%5E2%2B%F0%9D%91%80%2B1+%F0%9D%91%83%3D(%F0%9D%91%80%5E2%2B%F0%9D%91%80%2B1)%2F(%F0%9D%91%80%5E2%E2%88%92%F0%9D%91%80%E2%88%921)+It%27s+clear+from+the+last+derivation%2C+as+M+grows%2C+in+the+limit+P+will+tend+to+1%2C+because+the+values+of+M%2B1+and+M-1+would+be+negligible+compared+to+M%5E2+as+M+grows.+This+mean+%F0%9D%91%81%3D%F0%9D%91%83%F0%9D%91%80%5E2+has+no+positive+integer+solutions.+Was+Descartes+wrong+or+am+I+missing+something%3F&gs_lcrp=EgZjaHJvbWUyBggAEEUYOdIBCTIwMDdqMGoxNagCCLACAQ&sourceid=chrome&ie=UTF-8
Pay special attention to what step 2 says.
And then read attentively the final conclusion under the AI section "Solution".
This AI reviewed your writing even better that I could do it.
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About perfect numbers, and especially about odd perfect numbers, read this Wikipedia article
https://en.wikipedia.org/wiki/Perfect_number
You may continue your discussions with AI directly, passing me by.
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Hello, @math_tuitor2020, please remove all your insinuations
about me from all your posts - WITHOUT DELAY.
Jan. 20, 2025.
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Answer by math_tutor2020(3816)  (Show Source):
You can put this solution on YOUR website!
Of course ikleyn uses AI. Doesn't surprise me. Nothing of hers is original.
She should stop polluting this site with her garbage. And I found other solutions she completely plagiarized.
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