Here is one question which one of my students, Vedant Sahai asked me. It appeared in his computer subject exam of his recent ICSE X exam (Mumbai):
write a program to accept a number from the user, and check if the number is a happy number or not; and the program has to display a message accordingly:
A Happy Number is defined as follows: take a positive number and replace the number by the sum of the squares of its digits. Repeat the process until the number equals 1 (one). If the number ends with 1, then it is called a happy number.
For example: 31
Solution : 31 replaced by and 10 replaced by .
So, are you really happy? 🙂 🙂 🙂
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.
A prime number p is called a Wilson prime if . Using a computer and some programming language like C, C++, or Python find the three smallest Wilson primes.
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:
Amazon India link:
There are n points in a circle, all joined with line segments. Assume that no three (or more) segments intersect in the same point. How many regions inside the circle are formed in this way?
Do there exist 10,000 10-digit numbers divisible by 7, all of which can be obtained from one another by a re-ordering of their digits?
Solutions will be put up in a couple of days.