MHB Possibility of a Separate Forum for NT & Abstract Algebra

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The discussion advocates for the creation of separate forums for Number Theory (NT) and Abstract Algebra. Participants are exploring a mathematical problem involving the demonstration of infinitely many integers n such that both 6n + 1 and 6n - 1 are composite. One contributor proposes using n = 35x - 1 to derive the expressions for 6n + 1 and 6n - 1, successfully showing their composite nature. Another participant suggests an alternative approach with n = 36k^3, indicating a connection to the sum and difference of cubes. The conversation highlights both the need for specialized forums and ongoing mathematical exploration.
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1. There should be a separate (sub)forum for NT. ... and one for abstract algebra, for that matter!

2. Show that there are infinitely many n such that both 6n + 1 and 6n - 1 are composite. Without CRT, if possible.

My work... let n = 6^{2k}.
Then 6n \pm 1 = 6^{2k + 1} \pm 1...
Hmm. Having a hard timing finding the LaTexification button from my iPhone...
To be continued.
 
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Let n = 35x - 1, for any positive integer x.

6n + 1 = 6(35x - 1) + 1 = 210x - 5 = 5(42x - 1)
6n - 1 = 6(35x - 1) - 1 = 210x - 7 = 7(30x - 1)

QED

PS: there is no LaTeX yet
 
Bacterius said:
Let n = 35x - 1, for any positive integer x.

6n + 1 = 6(35x - 1) + 1 = 210x - 5 = 5(42x - 1)
6n - 1 = 6(35x - 1) - 1 = 210x - 7 = 7(30x - 1)

QED

PS: there is no LaTeX yet

Slick.
I also like n = 36k^3.
Then 6n \pm 1 is a sum/difference of cubes...
 

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