mersenneforum.org  

Go Back   mersenneforum.org > Extra Stuff > Miscellaneous Math

Reply
 
Thread Tools
Old 2022-04-11, 14:49   #12
charybdis
 
charybdis's Avatar
 
Apr 2020

85710 Posts
Default

Quote:
Originally Posted by sweety439 View Post
If we allow one of the three exponents be 2 (but of course cannot be 1), the other two exponents must be >= 3, do there exist infinitely many solutions other than 2^3+1^n=3^2?
No, by the Generalized Beal Conjecture as proved by Awojobi. Did you not read what I wrote??

The real answer is "yes". But you'll have to figure out for yourself why that is.

Last fiddled with by charybdis on 2022-04-11 at 14:51 Reason: oops
charybdis is offline   Reply With Quote
Old 2022-04-11, 15:32   #13
Dr Sardonicus
 
Dr Sardonicus's Avatar
 
Feb 2017
Nowhere

22×3×499 Posts
Default

Quote:
Originally Posted by sweety439 View Post
If we allow one of the three exponents be 2 (but of course cannot be 1), the other two exponents must be >= 3, do there exist infinitely many solutions other than 2^3+1^n=3^2?
There are infinitely many solutions to A^3 + B^3 = C^2 in positive integers A, B, C with gcd(A,B) == 1.

There are two types of solutions: those in which A + B and A^2 - A*B + B^2 are each 3 times a perfect square (e.g. 1^3 + 2^3 = 3^2) , and those with each a perfect square (e.g. 56^3 + 65^3 = 671^2).

I leave it as an exercise to work out formulas for each case.
Dr Sardonicus is offline   Reply With Quote
Old 2022-04-12, 04:40   #14
sweety439
 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

72×73 Posts
Default

Does there exist a positive integer N, such that if the sum of the three exponents is >=N (and none of the three exponents is 1, and at most one of the three exponents is 2), then there exist only finitely many solutions other than 2^3+1^n=3^2? If so, find the smallest such positive integer N

(I also think that 2^3+1^n=3^2 will be the only solution if N is enough large)

Last fiddled with by sweety439 on 2022-04-12 at 04:42
sweety439 is offline   Reply With Quote
Old 2022-04-12, 04:56   #15
sweety439
 
"99(4^34019)99 palind"
Nov 2016
(P^81993)SZ base 36

67718 Posts
Default

Quote:
Originally Posted by Dr Sardonicus View Post
There are infinitely many solutions to A^3 + B^3 = C^2 in positive integers A, B, C with gcd(A,B) == 1.

There are two types of solutions: those in which A + B and A^2 - A*B + B^2 are each 3 times a perfect square (e.g. 1^3 + 2^3 = 3^2) , and those with each a perfect square (e.g. 56^3 + 65^3 = 671^2).

I leave it as an exercise to work out formulas for each case.
I found that sequence is OEIS: A099426, also numbers n such that n^2 is in A202679
sweety439 is offline   Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
Can we prove Beal conjecture assuming ABC conjecture? didgogns Miscellaneous Math 1 2020-08-05 06:51
"PROOF" OF BEAL'S CONJECTURE & FERMAT'S LAST THEOREM Awojobi Miscellaneous Math 50 2019-11-02 16:50
The Beal Conjecture Proof Arxenar Miscellaneous Math 1 2013-09-07 09:59
Distributed Beal Conjecture Problem Joshua2 Math 54 2009-10-19 02:21
New Beal Conjecture Search Joshua2 Open Projects 0 2009-04-20 06:58

All times are UTC. The time now is 05:41.


Sat Oct 1 05:41:46 UTC 2022 up 44 days, 3:10, 0 users, load averages: 1.20, 1.30, 1.33

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2022, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.

≠ ± ∓ ÷ × · − √ ‰ ⊗ ⊕ ⊖ ⊘ ⊙ ≤ ≥ ≦ ≧ ≨ ≩ ≺ ≻ ≼ ≽ ⊏ ⊐ ⊑ ⊒ ² ³ °
∠ ∟ ° ≅ ~ ‖ ⟂ ⫛
≡ ≜ ≈ ∝ ∞ ≪ ≫ ⌊⌋ ⌈⌉ ∘ ∏ ∐ ∑ ∧ ∨ ∩ ∪ ⨀ ⊕ ⊗ 𝖕 𝖖 𝖗 ⊲ ⊳
∅ ∖ ∁ ↦ ↣ ∩ ∪ ⊆ ⊂ ⊄ ⊊ ⊇ ⊃ ⊅ ⊋ ⊖ ∈ ∉ ∋ ∌ ℕ ℤ ℚ ℝ ℂ ℵ ℶ ℷ ℸ 𝓟
¬ ∨ ∧ ⊕ → ← ⇒ ⇐ ⇔ ∀ ∃ ∄ ∴ ∵ ⊤ ⊥ ⊢ ⊨ ⫤ ⊣ … ⋯ ⋮ ⋰ ⋱
∫ ∬ ∭ ∮ ∯ ∰ ∇ ∆ δ ∂ ℱ ℒ ℓ
𝛢𝛼 𝛣𝛽 𝛤𝛾 𝛥𝛿 𝛦𝜀𝜖 𝛧𝜁 𝛨𝜂 𝛩𝜃𝜗 𝛪𝜄 𝛫𝜅 𝛬𝜆 𝛭𝜇 𝛮𝜈 𝛯𝜉 𝛰𝜊 𝛱𝜋 𝛲𝜌 𝛴𝜎𝜍 𝛵𝜏 𝛶𝜐 𝛷𝜙𝜑 𝛸𝜒 𝛹𝜓 𝛺𝜔