Let a and b satisfy and .
- Prove that if m and n are positive integers with , then .
- For each positive integer n, consider a quadratic function: .
Show that has two roots that are in between -1 and 1.
Let . Consider with . Since , we have . Hence, . Call this relationship I.
On the other hand, notice that , , , , which implies
….call this relationship II.
From relationships I and II, it follows that
which can be written as
, or equivalently, . That is, .
It remains to prove that . Indeed, as .
The equality occurs if and only if .
2) Since discriminant , has two distinct real roots . Also, note that if , then the following holds:
We conclude that .
Reference: Selected Problems of the Vietnamese Mathematical Olympiad (1962-2009), Le Hai Chau, Le Hai Khoi.
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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.