maths

i like maths
numbers and such, that whole thing
i'll put numerical/mathematical thoughts here

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contents of page

• all mersenne numbers with composite exponents are composite proof (published 2022-10-05)
• 2+2=5 proof (published 2021-11-18)

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all mersenne numbers with composite exponents are composite proof

this has been proven before but i made this proof independently

\(x^2 - 1\) can be factored into \((x-1)(x+1)\). (difference of two squares)

\(x^3 - 1\) can be factored into \((x-1)(x^2+x+1)\). (difference of two cubes)

Extending on this, \(x^4 - 1\) can be factored into \((x-1)(x^3+x^2+x+1)\).

In general, \(x^y - 1\) can be factored into \(\displaystyle(x-1) \sum_{n = 0}^{y-1} x^n\).

A Mersenne number with a composite exponent can be represented as \(2^{ab} - 1\), where \(a\) and \(b\) are integers greater than 1.

This can be rewritten as \((2^b)^a - 1\), an \(a\)-th power minus 1.

As \(x^y - 1\) can be factored into \(\displaystyle(x-1) \sum_{n = 0}^{y-1} x^n\), \((2^b)^a - 1\) can be factored into \(\displaystyle(2^b-1) \sum_{n = 0}^{y-1} (2^a)^n\), meaning \(2^b - 1\) is a factor of \(2^{ab} - 1\).

As \(b\) is an integer greater than 1, \(2^b - 1\) is an integer greater than 1, and therefore makes \(2^{ab} - 1\) composite. \(\blacksquare\)

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2+2=5 proof

this is a proof that i made in 2018 that 2+2 = 5
the proof is copied verbatim from the original

A proof that 2+2=5.

Let a=b.
Multiply by a on both sides to make a^2 = ab.
Add a^2 on both sides to make 2a^2 = a^2 + ab
Subtract 2ab on both sides to make 2a^2 - 2ab = a^2 + ab - 2ab
Simplify to 2a^2 - 2ab = a^2 - ab

Add 3a^2 - 3ab to both sides to make 5a^2 - 5ab = 4a^2 - 4ab

Take the 4 out of the right side to make 5a^2 - 5ab = 4(a^2 - ab)

Change the 4 to 2+2 to make 5a^2 - 5ab = (2+2)(a^2 - ab)

Take the 5 out of the left side to make 5(a^2 - ab) = (2+2)(a^2 - ab)

Divide by a^2 - ab to get 5 = 2+2

Rearrange to get 2+2 = 5.

i didnt write down what the error in the proof was originally but the error in the proof is (hover over spoiler to reveal) in the second last line i divide by a^2 - ab, but if a=b as said in the start, a^2 - ab = a^2 - a^2 = 0; i divided by 0, producing the incorrect answer of 2+2=5.

update on 2022-10-05:

i just learned LaTeX so here it is presented in LaTeX

$$a=b$$ $$a^2 = ab$$ $$2a^2 = a^2 + ab$$ $$2a^2 - 2ab = a^2 + ab - 2ab$$ $$2a^2 - 2ab = a^2 - ab$$ $$5a^2 - 5ab = 4a^2 - 4ab$$ $$5a^2 - 5ab = 4(a^2 - ab)$$ $$5a^2 - 5ab = (2+2)(a^2 - ab)$$ $$5(a^2 - ab) = (2+2)(a^2 - ab)$$ $$5 = 2+2$$ $$2+2 = 5$$