Ramanujan summation

In 1913, Ramanujan wrote this to G.H Hardy:
"Dear Sir, I am very much gratified on perusing your letter of the 8th February 1913. I was expecting a reply from you similar to the one which a Mathematics Professor at London wrote asking me to study carefully Bromwich's Infinite Series and not fall into the pitfalls of divergent series. … I told him that the sum of an infinite number of terms of the series: 1 + 2 + 3 + 4 + · · · = −1/12 under my theory. If I tell you this you will at once point out to me the lunatic asylum as my goal. I dilate on this simply to convince you that you will not be able to follow my methods of proof if I indicate the lines on which I proceed in a single letter. …"
This is a hint of what he meant:

Thomas Jefferson: Cryptographer

The Wheel Cipher


Thomas Jefferson, already absurdly accomplished by 1795, somehow found time to delve into cryptography, where he devised this cipher system. The letters of the alphabet are printed along the rim of each of 36 disks, which are stacked on an axle. One party rotates the disks until his message can be read along one of the 26 rows of letters, somewhat like a modern cylindrical bike lock. Now he can record the letters in any one of the other 25 rows and send that string safely to another party, who decodes it by reversing this procedure. If the message is intercepted, it’s useless even to someone who has the disks, because he must also know the order in which to stack them, and 36 disks can be stacked in 371,993,326,789,901,217,467,999,448,150,835, 200,000,000 different ways.

This is pretty robust. The cipher below, created in 1915 by U.S. Army cryptographer Joseph Mauborgne, has never been solved. “The known systems from this year (or earlier) shouldn’t be too hard to crack with modern attacks and technology,” writes NSA cryptologist Craig P. Bauer. “So, why don’t we have a plaintext yet? My best guess is that it used a cipher wheel” like Jefferson’s.

mauborgne cipher

(L. Kruh, “A 77-year-old challenge cipher,” Cryptologia, 17(2), 172-174, 1993, quoted in Bauer’sSecret History: The Story of Cryptology, 2013.)

Some Things You Just Have to Know

The 'fact' about the stars in the sky being so far way...  Is flat out incorrect!  These are Milky Way galaxy stars (where we live) and they aren't that far away.  Most stars live millions to many billions of years.  The furthest star you are going to see, with your eyes, are, at most, ~10000 light years distant.  That's not very long in star time!

The most stars you can see with your naked eye is about ten thousand of them...roughly the number of grains in a handful of sand.  This is only the smallest fraction of the number of stars that are.  The number of stars that are  is greater than all the grains of sand on all the beaches on earth.  - Carl Sagan


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