Tribute to Ronald Graham's Largest Number ...262464195387

What is the largest number? Any number that is offered can be bettered by that number plus 1. A common answer is infinity, although infinity isn't a specific number. "Infinite" describes something that is without bounds. Something could be infinitely large (set of integers) or something could be into infinitesimally small (as done in calculus). 

Another approach to this question of the largest number is to ask what is the largest number used in a proof. In 1977 the mathematician Ronald Graham established the world record for the largest specific integer used in a mathematical proof. Graham's number is so large that if all the atoms in the universe were made into ink, the number could not be written. That is certainly disappointed for the readers of this post who wanted to see the number.

However, the last digits of Graham's number are: ...262464195387.

The actual expression of Graham's number uses hyperoperators which are higher order forms of exponentiation. 

Since 1977, Graham's number has been exceeded by larger super-numbers used in proofs. One example is TREE(3).

Ronald Graham passed away in 2020 and was honored in a Numberphile podcast. An earlier podcast specifically described Graham's number. While the end digits of Graham's number have been determined, the first digit is unknown. Graham was asked what digit he would like it to be and he said he actually knew the first digit but only when the number is expressed in base-2 and then it is 1.

The icon I chose for this post is juggling because Ron Graham was an accomplished juggler. For a photo of Graham, see New York Times Obituary Link with photo.

A Simple Pythagorean Theorem Proof

 

I recently came across a proof for the Pythagorean Theorem that is simple and contains a minimum amount of equations. It may be one of the most direct and understandable proofs of the many that exist. A nice animation of this proof is found at MathAdam.

 

Start with your standard right-angle triangle of sides A, B and C with C being the hypotenuse.

 

Draw a line perpendicular to C to the opposite vertex.

Now, in addition to the original triangle, there are two additional right-angle triangles formed. One has the side A as its hypotenuse (well name triangle A) and the other has side B has its hypotenuse (triangle B). The original triangle with sides ABC will now be called triangle C. The three triangles, A, B, and C are all similar A ~ B ~ C. (This can be established by the fact all three has the same interior angles.) Another key fact that we can see is the area of A + B =  area of C

The next step is to flip over each of the three triangles.

The next step is to the draw out squares on each of the three sides.

The relative sizes of the squares have the same relative ratios as the triangles A:B:C.

Since the areas of triangles has the relationship: A + B = C, then the same applies for the squares, therefore A² + B² = C².

                

One-Time Pad Coding and a Proposal for Improvement

(Image: Iconfinder - Font Awesome on Iconfinder)

A one-time pad (OTP) is a coding system that cannot be cracked. The system uses a set of random characters (the OTP) to encode a message by a sender. The receiver, who also has the OTP, decodes the message. The rules for keeping the OTP secure are:

1. Only the sender and receiver have the OTP. That is, the key is truly secret.

2. The OTP is only used once.

3. The OTP must be at least the same length as the message. If it isn't, then part of the OTP would be used over and that would violate Rule 2 above.

4. The OTP must be truly random.

The term one-time pad is used because the original system used two identical pads of sheets containing random characters. The users would each use one sheet from their pads for a message then destroy the sheets used. 

It has been proven that the OTP provides perfect secrecy. This seems like a bold statement given the ever-increasing power of computers. One would think the code could be cracked by use of the brute force method - using a super-computer to try every possible key to decrypt the encoded message. While a super-computer could try every possible key, in doing so it would also produce every possible message of the given length of the OTP.

Here's an example, I used the Boxentriq Code-Breaking site to encode the ten-character message "KISS POTUS" using the OTP key of "JIHGFEDCBA" to produce the result "TQSRTRVVS". My key doesn't follow Rule 4 but it I'm using it just for demonstration purposes. The supercomputer would eventually get to the key "JIHGFEDCBA" and produce the result "KISS POTUS"; however, it would also produce every other possible ten-character combination, including "KICK POTUS", "I LIKE YOU", "GO HOME NOW" and millions of other possible messages and many more millions of just random, scrambled letters. While the supercomputer "cracked" the code, one would not know which of the millions of results was the intended message.

There are weaknesses of the OTP system. Keeping the key secret is a human weakness. Producing truly random characters is system weakness. 

A Proposal for Improvement

Instead of the sender and receiver having long, identical pads, they agree on the digits of a single irrational number to be used as the key for the OTP. Irrational numbers, such as the square root of 2, can be expressed succinctly (and have the advantage that their digital expansion continues infinitely without any recognizable pattern - square root of 2 = 1.414213562373095048801688724210...).

The users of the code would agree on what irrational number to be the key for their initial message. The sender would reserve the last few characters of the message to indicate the key for the next message - another irrational number. There is an unlimited supply of such numbers. One could also have a program to randomly produce each subsequent key to use. 

I'm new to this subject, so it is very possible someone has already suggested this method. 

A site with resources for many code systems: Boxentriq Code-Breaking 

Wikipedia entry: One-time pad

An example of a ten million digit pad based on the digits of pi. Digits of pi are discussed in these other posts.

8/6/2021 update: also review Vigenere cipher

6/21/2024 updated: I've made a Mathematica routine to encode and decode messages using the strategy noted in this blog post. Leave a comment with your email and I'll share with you.