Math Musings

Probability theory  is a branch of mathematics concerned with the analysis of random phenomena. The outcome of a random event cannot be determined before it occurs, but it may be any one of several possible outcomes. The actual outcome is considered to be determined by chance. It is increasingly making its way into all branches of mathematics including number theory where at times mathematicians are looking for answers in a statistical sense. Here is an innocuous-sounding probability question that is not as easy as it sounds. What is the expected number of tosses of an unfair coin needed to get two heads in a row (assume probability p of a head)? Same question for three heads in a row and j heads in a row.  Start with a simple case First: What if you only need the expected number of tosses required to get one head? Let N be the number of coin tosses, then we want to find E(N|1H) (expected number of tosses given that you seek only one head). Toss the coin once. Either you g...

  Two distinct positive integers a and b are said to form an amicable pair if the sum of the proper divisors of a is equal to b , and the sum of the proper divisors of b is equal to a . A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1,2, and 3. The ancient Greeks were already interested in amicable pairs, and they knew of an example: 220 and 284. The proper divisors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, and 110. The proper divisors of 284 are 1, 2, 4, 71, and 142. If we write s(n) for the sum of the proper divisors of n. We have 220 and 284 was the only amicable pair known to the ancient Greeks, but over the centuries, two more pairs were found: 17296 and 18416, and 9363584 and 9437056. Those three pairs were all that were known before Euler tackled the problem of finding amicable pairs, and he found 58 more pairs, thus increasing the supply of known pairs from 3 all the way up to...

  The solution to the equation    has been recently discovered. A much-studied equation that went unsolved for 64 years. Andrew Booker, a mathematician at the University of Bristol, has finally solved it; He discovered that Booker found this odd trio of 16-digit integers by devising a new search algorithm to shift them out of quadrillions of possibilities. The algorithm ran on a university supercomputer for three weeks straight. (He says he thought it would take six months, but a solution "popped out before I expected it") The field of number theory deals with finding integer solutions to equations. This is a case where finding the solution with pen and paper would have been incredibly challenging. So, a super computer came to the rescue. Perhaps one can argue that for some problem’s computers can do a better job. Nevertheless, they lack imagination -a crucial ingredient – and thus the human mind still has a considerable edge for the vast majority of hard problems tha...

I recently gave a talk on Internet Security and Cryptography at the international CMSC conference on Jul 3, 2020. Kindly see the enclosed video for the 5 minute video explaining the essence of Elliptic Curve Cryptography. 

Prof Parimala Raman is the only lady mathematician educated in India who is currently member of all three Indian Academy of Sciences besides winning the prestigious Shanti Swarup Bhatnagar award in 1987. In 2010, she was selected as the plenary speaker at the International Congress of   mathematicians. When the government of India decided to establish eleven chairs to honor women in math and sciences recently, Prof Raman is the only living person in the list. Her early years were spent in Chennai where the love and support of her parents and teachers got her interested in math. In her undergraduate, she seriously considered majoring in Sanskrit poetry but the mathematician won that battle. It is not surprising both the fields require a lot of originality and creativity. She went on to get her doctorate at the prestigious Tata Institute of Fundamental research (TIFR). She always had excellent mentors in her early years of research and that was the sounding board whi...

A ring is a group with additional features in abstract algebra. But first let’s look at the set of integers, set of real numbers, A matrix with 2 rows and 3 columns and the set of complex numbers. If we look at the set of integers(Z), it is closed under addition, subtraction and multiplication meaning that if we perform these operations on two integers, we will get another integer. However, if we divide two integers, we might not get another integer but rather a fraction. In a similar vein, the set of real numbers(R) will be closed on all these four operations provided we do not divide by 0. In the case of  R 2x3   matrix, while it will be closed under addition and subtraction, we cannot multiply two matrices of arbitrary sizes but rather follow the rule and inverse of a matrix does not always exist.   Lastly, if we take two polynomials and divide them, we get a rational function so the set of complex numbers is not closed under division. We alr...

Livingston High School student Rohan Jha won a prize of "200 Pi Dollars." See what that comes out to in U.S. currency here. LIVINGSTON, NJ — For a glimpse into the tongue-in-cheek tone of the Steven H. Strogatz Prize for Math Communication, one needs only look at the prize of "200 Pi Dollars." For those not mathematically inclined, that total – which comes out to about $628 (200 x 3.14159 = $628.32) – was what Livingston high school student Rohan Jha earned via the inaugural award event. The contest was spearheaded by the National Museum of Mathematics (MoMath) in New York City. It seeks to highlight high school students who "celebrate the universality of math" using social media, visual art, writing and dance. The 2020 winners, their projects and the original call for entries can be viewed here. "The purpose of Math Musings, the magazine I started in high school, was to show that math is everywhere, yet many times we are not awar...