The regular pentagram was the badge of the Pythogoreans.But the Pythagoreans first thought that all the rations were rational,i.e,\(x = \frac{a}{b}\). where a nd b are rational numbers.

Clearly this fraction must follow the equation for the golden ratio(It's a property of the pentagram),and hence we get \[a^{2} = ab+b^{2}\]

If \(a\) and \(b\) are of different parity clearly this leads to a contradiction.Also the assumption that \(a\) and \(b\) are odd also leads to a contradiction.

Hence both \(a\) and \(b\) must be even,that is \(a = 2a_1\) and \(b = 2b_1\),where \(a_1 < a\) and \(b_1 < b\).

Substituting this into the equation and cancelling we get \[a_1^{2} = a_1b_1 + b_1^{2}\].

Now the same logic may be applied to this equation also which gives \(a_1 = 2a_2\) and \(b_1 = 2b_2\) where \(a_2 <a_1\) and \(b_2 < b_1\).

So we obtain an infinitely decreasing sequence of positive integers,ie \(a>a_1>a_2>a_3>.....\) and \(b>b_1>b_2>b_3>...........\).

Clearly no such infinitely decreasing sequence exists for positive integers.

For more details on this approach you may choose to read my note on Fermat's Method of Infinite Descent

## Comments

Sort by:

TopNewestNice note! But in my opinion, the note jumps out on you, does its job and then runs away.

Additional background, like what the golden ratio is, what a pentagram is and how these two are related would have been helpful.

This note proves that golden ratio can not be expressed in the form \(\frac{a}{b}\) where \(a\) and \(b\) are positive integers. But it does not

explicitlymentions this statementanywhere.Despite all this, this is a nice demonstration of proofs that use the method infinite descent. – Mursalin Habib · 3 years, 1 month ago

Log in to reply

Stochastic Programming.It will be extremely helpful for me if you write a small review of that note too.....I put a lot of effort into this one but it didn't get attention.Thanks in advance. – Eddie The Head · 3 years, 1 month ago

Thanks for the constructive review.Actually I didn't include the golden ration in this note because many people have posted about it before. Here is my note onLog in to reply

can we say that infinite descent is a strong form of pentagram? – Niladri Dan · 3 years, 1 month ago

Log in to reply

– Eddie The Head · 3 years, 1 month ago

The symbol is the pentagram and infinite descent is the proof technique....clck on the link if you're interested to learn more about it...Log in to reply