The eminent British mathematician had once remarked: Every integer was a friend to Srinivasa Ramanujan.
Well, we are mere mortals, yet we can cultivate some “friendships with some numbers”. Let’s try:
Squaring 12 gives 144. By reversing the digits of 144, we notice that 441 is also a perfect square. Using C, C++, or python, write a program to find all those integers m, such that , verifying this property.
PS: in order to write some simpler version of the algorithm, start playing with small, particular values of N.
Reference: 1001 Problems in Classical Number Theory, Indian Edition, AMS (American Mathematical Society), Jean-Marie De Konick and Armel Mercier.
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It is quite well-known that any positive integer can be factored into a product of primes in a unique way, up to an order. (And, that 1 is neither prime nor composite) —- we all know this from our high school practice of “tree-method” of prime factorization, and related stuff like Sieve of Eratosthenes. But, it is so obvious, and so why it call it a theorem, that too “fundamental” and yet it seems it does not require a proof. It was none other than the prince of mathematicians of yore, Carl Friedrich Gauss, who had written a proof to it. It DOES require a proof — there are some counter example(s). Below is one, which I culled for my students:
(a) Show that the sum and product of elements of E are in E.
(b) Define the norm of an element by . We say that an element is prime if it is impossible to write with , and , ; we say that it is composite if it is not prime. Show that in E, 3 is a prime number and 29 is a composite number.
(c) Show that the factorization of 9 in E is not unique.
Show that for each positive integer n equal to twice a triangular number, the corresponding expression represents an integer.
Let n be such an integer, then there exists a positive integer m such that . We then have so that we have successively
; ; and so on. It follows that
, as required.
Comment: you have to be a bit aware of properties of triangular numbers.
1001 Problems in Classical Number Theory by Jean-Marie De Koninck and Armel Mercier, AMS (American Mathematical Society), Indian Edition:
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Most math concepts are intuitive, simple, yet subtle. A similar opinion is expressed by Prof. Michael Spivak in his magnum opus, Differential Geometry (preface). It also reminds me — a famous quote of the ever-quotable Albert Einstein: “everything should be as simple as possible, and not simpler.”
I have an illustrative example of this opinion(s) here:
Consider the principle of mathematical induction:
Most students use the first version of it quite mechanically. But, is it really so? You can think about the following simple intuitive argument which when formalized becomes the principle of mathematical induction:
Theorem: First Principle of Finite Induction:
Let S be a set of positive integers with the following properties:
- The integer 1 belongs to S.
- Whenever the integer k is in S, the next integer must also be in S.
Then, S is the set of all positive integers.
The proof of condition 1 is called basis step for the induction. The proof of 2 is called the induction step. The assumptions made in carrying out the induction step are known as induction hypotheses. The induction situation has been likened to an infinite row of dominoes all standing on edge and arranged in such a way that when one falls it knocks down the next in line. If either no domino is pushed over (that is, there is no basis for the induction), or if the spacing is too large (that is, the induction step fails), then the complete line will not fall.
So, also remember that the validity of the induction step does not necessarily depend on the truth of the statement that one is endeavouring to prove.
Hidden truths permeate our world; they’re inaccessible to our senses, but math allows us to go beyond our intuition to uncover their mysteries. In this survey of mathematical breakthroughs, Fields Medal winner Cédric Villani speaks to the thrill of discovery and details the sometimes perplexing life of a mathematician. “Beautiful mathematical explanations are not only for our pleasure,” he says. “They change our vision of the world.”
Cheers, cheers, cheers,
thanks Dr. Villani,
This is a lucid, lovely, condensed lecture on a brief history of primes by Prof. Manindra Agrawal, Department of Computer Science, IIT Kanpur delivered under the auspices of Clay Math Institute some years before.
I am sure you will like it 🙂 🙂 🙂