Just a few minutes ago I thought that may be the next title to this new post would also be funny: WTF! Doing math research with the help of a Google search suggestion??? And yes, if you think about that it is rather weird but it is true. How come I have fallen so deep?
Well a couple of weeks back I did a Google internet search by typing in one of my counter examples to the last theorem of Pierre de Fermat. In the normal html search results nothing popped up but when I looked in the pictures to my surprise every time there was only one picture of my website. And indeed that picture contained the search phrase I had just typed in… So at present date & year the Google search algorithm can translate a phrase like 5^n + 7^n = 12^n mod 35 to the actual content of a picture on this website. By all standards this is amazing. Here are the two search phrases I did put in and if you do that yourself you can find back in the pictures from Google search the actual picture on this website that the search phrase is about. Here are the two pictures explaining the stuff a little bit:
When I started this website in 2015 in the second post I explained how this website was set up: The math will be mostly in the pictures I create so it is a pity that internet search engines cannot read that. Well now almost six years later as a matter of routine the Google search engine can indeed a bit of the math content as found in my pictures.
Here is the second search string or search phrase:
To be honest I was amazed at the results. So later I decided to check what would pop up on Google if I would search for my new little theorem of Pierre de Fermat. So my search phrase was p^q = p mod pq and yes Google can find that content in my pictures too.
But the Google search engine also popped up an alternative search phrase also containing the mod pq stuff. And for a few seconds of time I started to panic like ‘Oh oh have the math professional professors already found the new little theorem of Fermat and am I only an idiot doing weird stuff that is known to science for centuries????‘ But very soon I calmed down, ok they might have proven that identity that Google search suggests, but they are still overpaid weirdo’s never ever able to connect this particular identity to the new little theorem of Fermat. Compare it to the Cauchy-Riemann equation that rule differentiation in the Complex plane; despite having those 2D equations the math professors just cannot bring those same ideas to 3D space. And not for just a few years or so, no they cannot do that century in century out. Even if there is a dedicated website out like this website doing a lot of 3D complex number theory, the overpaid & incompetent math professors keep on being silent year in year out and likely also century in and century out.
But let’s not get emotional about how stupid this all is, after the indentity found is very beautiful and all in all very much like my new little theorem of Pierre de Fermat. And may be it is a bit unwise from me to view the collective of math professors as ‘utterly stupid’ After all they have some kind of proof for it and most of those proofs use the CRT or the Chinese remainder theorem. And that made me pause for a moment, the remainder theorem is very old so how the hell did China industrial spionage in other parts of the world in those long lost centuries? Did they find the CRT without stealing intellectual property? Wow!
But let’s get serious, this post is five pictures long and all about that search suggestion as done by Google. Before we start I once more want to show you that the new little theorem of Fermat is indeed an improvement. For those who already know that, just skip it if you want.
The old theorem of Fermat says that a^p = a mod p, here p is a prime number and p should not be a factor of a. So a and p must be relatively prime to each other. But if p is a small number, it does not work that well. Take for example a = 105 and p = 2. Well 105^2 mod 2 = 1 but everybody already knows that the square of an odd number is odd.
The new little theorem of Fermat says a^p = a mod ap and because ap is always greater than a, this also returns a when p is small. For example now we have 105^2 mod 210 = 105. So I think this is a small improvement.
Ok, let us go to the five pictures:
Yes it is the end of this post but I would like to remark that if you look at the banner of this website it says ‘Math as you have never seen it before’. Of course that is not a goal an sich but a natural by product of the things I like to study. My math all hang together in ways that are often not obvious. For example the new little theorem of Fermat is not a stand alone result, it emerges directly from when I studied the exponential orbits during all those counter examples to the last theorem of Fermat. In return all those counter examples started with the first ones I found in my own 3D complex numbers. So this all hangs together while understanding this new little Fermat theorem can also be understood using just plain old math without any 3D complex numbers or so.
The professional math professors never found this new little Fermat theorem because they do not want to walk the path from 3D complex numbers to where we are now. They only want the 2D complex plane and ok may be the cute quaternions and that’s it. Of course that is not forbidden, but if in 3.5 centuries of time you even cannot find the new Fermat little theorem, might it be time to scratch the head a little bit and do some rethinking?
Of course not my dear reader, the math professors will never change.
See you in the next post & thanks for your attention.