The Man Who Loved Only Numbers.

 

The man who loved only numbers – by Paul Hoffman

The book is a biography of Paul Erdos, a Hungarian mathematician.

Paul was highly talented and his talent was realized by his parents when they saw him doing three digit multiplications in his mind at the age of three. He could quickly convert the ages of strangers from years to seconds in his mind. When he was 4 years he understood negative numbers and interestingly from that moment he was known to spread lot of positivity around him.

The positivity was seen in his work for years. Erdos was not a paranoid like most others. He believed that team work gives the best results. He embraced a transient lifestyle, living out of a suitcase and constantly traveling to visit his colleagues all over the world, earning him the nickname "The Peripatetic Mathematician."

Keep your SF score low:

Throughout his studies, Erdos was smitten with prime numbers, and as a Jew, he not only believed in God, but he strongly conjectured that God (whom he called the Supreme Fascist, or SF), possessed a “transfinite book,” which held the key to all mathematical problems in the existence of the universe. The Erdos often said that the “Game of life” is to keep the SF score low. If you do something bad in life, the SF gets two points. If you don’t do something good that you should have done the SF gets one point. You never score so the “SF always wins.”

The way we find limitations in dogs when they don’t understand Quadratic equations, simply because their brain is not made for it, the SF laughs at us when we can’t solve certain problems. Nevertheless in 1932 he became internationally famous at his age of 20 when he got a simple proof of a theorem that was originally conjectured by Bertrand and later proved by Tchebychev: For every positive integer n greater than 1, there is a prime between n and 2n.  Erdos and Pomerance did another important work on prime numbers. If n is a prime number, then for every integer a, the number (an –a) should be a multiple of n.

A century ago Euler had advanced a formula n2 + n + 17, which for successive values of n gives a prime number. It was seen working from n = 0 to n = 15, but did not work for n =16. It gave 289 which isn’t prime. Euler modified it to n2 + n + 41 which worked for integers from 0 to 39.

Math needs optimism:

A Polish-born mathematician Stanislaw Ulam recommends a “I can do it” mindset for anyone to enjoy Math. Being optimistic it helps the process of problem solving to be long and that’s the key. On the other hand the pessimistic attitude increases the fatigue that engenders the self-doubt.

Math also expects creativity. One of the intern working with Paul, decided to quit the life loaded with Math. Paul asked him, “What have you decided to do?” He replied, “I want to be Poet.” Paul’s retort was, “That’s good decision. Math needs deep creativity, more than that needed to be poet.”

 Moreover the mind of a mathematician operates in a unique way, often obsessively seeking patterns and connections in the most abstract concepts. An example of Erdős's obsessive search for patterns can be seen in his work on the Collatz conjecture, also known as the 3n+1 problem. This conjecture involves a series of manipulations on a positive integer: if the number is even, divide it by 2; if it is odd, multiply it by 3 and add 1. The sequence is then repeated until it eventually reaches the number 1.

Prime numbers can’t be easily tamed. (Mersenne number)

In 1742, Christian Goldbach conjectured that every even number greater than 2 is the sum of two prime numbers.

                                        4 = 2 + 2

                                        6 = 3 + 3

                                        8 = 3 + 5

                                        10 = 5 + 5

                                        12 = 5 + 7

                                        14 = 7 + 7….and  so on.

In the seventeenth century, a Parsian monk found 2n – 1 as prime number, where n remains a prime. For the fifth prime number it didn’t work. 211 – 1= 2047. This number has 23 and 89 as its factors.

 Ramsey theory:

https://youtu.be/-CxDfy7AsV8?si=bnOUXYZyTS0CJcu7

 Complete disorder is impossible.

What’s the smallest number of people so that there must be 3 mutual friends or 3 mutual strangers?

The answer is 6.

Consider 5 friends A, B, C, D and E.

Mutual friends             : AB, BC, CD, DE and AE

Mutual strangers          : AC, AD, BD, BE, CE

A and B are friends, B and C are friends but A, B and C are not mutual friends since A and C are strangers. It doesn’t happen with five or less friends.

 Now consider 6 friends A, B, C, D, E and F.

What’s the smallest number of people so that there must be 4 mutual friends or 4 mutual strangers?

The answer is 18.

What’s the smallest number of people so that there must be 5 mutual friends or 5 mutual strangers?

The mathematicians don’t know. All that is known is that the answer is between 43and 48, both inclusive.

 Another example of Ramsey theory:

If you take first (n2 + 1) integers to arrange in any order. No matter how perverse the arrangement, you will always be able to find n + 1 integers that form an increasing sequence or a decreasing sequence. With n2 integers you may not get it.

For eg. Take first 25 integers, we will always find 6 integers that form increasing or a decreasing sequence. To prove it wrong, let’s restrict our arrangement to 5.

21, 22, 23, 24, 25, 16, 17, 18, 19, 20, 11, 12, 13,1 4, 15, 6, 7, 8, 9, 10, 1, 2, 3, 4, 5.

We’ve restricted the longest chain to be 5. Now if we add the number 26 anywhere we want we are bound to find a six-term sequence with all the numbers rising or all falling.

 Unit fractions, the love of Egyptians.

The Egyptians dealt a lot with unit fractions. Unit fractions are reciprocals of positive integers, like 1/5, 1/8, 1/127. These are the fractions in which the numerators is 1. Mr. Fibonacci preferred to construct unit fractions by so called greedy procedure, in which largest possible unit fraction less is chosen for each term of the expansion. The greedy expansion of 3/7 for instance, yields

3/7 =  (1/3) + (1/11) + (1/231)

Take largest unit fraction less than (3/7). It is (1/3)

How?

(3/7) = (1/2.33) So (1/2) will be a shade bigger than (3/7) and (1/3) will be the largest unit fraction less than (3/7).  Now subtract (1/3) from (3/7) to get (2/21). Again the largest unit fraction less than (2/21) = (1/11). Subtracting (1/11) from (2/21) we get (1/231)

Fibonacci showed that this greedy procedure always produces a sum of fractions that terminates.

2/5 = (1/3) + (1/15)

2/7 = (1/4) + (1/28)

2/13 = (1/7) + (1/91)  

2/15 = (1/8) + (1/120)   

Disintegration of fractions:

An ordinary fraction can be expressed as a sum of unit fractions in infinitely many ways using the identity

1/a = [1/(a+1)] + [1/a(a+1)]

1/2 = 1/(2+1) + [1/2(2+1)]  = (1/3) + (1/6)….(1)

1/3 = [1/(3+1)] + [1/3(3+1)] = (1/4) + (1/12)….(2)

Putting (2) in (1) we get 1/2 = (1/4) + (1/12) + (1/6)  = (1/2)

Applying one more time…(1/4) = (1/5) + (1/20)

So 1/2 = (1/5) + (1/20) + (1/12) + (1/6)  = (1/2)..This can go on and you can do this to your heart’s content.

Erdos old age:

https://youtu.be/EGc6rE24YSw?si=A0AxKVm_6wIGGgdi

 Erdos puts it, quoting his friend Ulam, “ The first sign of senility is that a man forgets his theorems, the second sign of senility is that he forgets to Zip up and the third sign of senility is that he forgets to Zip down.”

https://www.google.com/gasearch?q=paul%20erdos&tbm=&source=sh/x/gs/m2/5

 

Disclaimer: The summary of this book is just an attempt to collect the points that I feel are too good to forget. In the process of making these points I have used my comfort, my examples and modifications.

 

Vinay Wagh

Bulls Eye

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