Explain why the following calculations hold:
Explain why the following calculations hold:
I would strongly suggest to read the book “Men of Mathematics” by E. T. Bell.
It helps if you start at a young age. It doesn’t matter if you start later because time is relative!! 🙂
Well, I would recommend you start tinkering with mathematics by playing with nuggets of number theory, and later delving into number theory. An accessible way for anyone is “A Friendly Introduction to Number Theory” by Joseph H. Silverman. It includes some programming exercises also, which is sheer fun.
One of the other ways I motivate myself is to find out biographical or autobiographical sketches of mathematicians, including number theorists, of course. In this, the internet is an extremely useful information tool for anyone willing to learn…
Below is a list of some famous number theorists, and then there is a list of perhaps, not so famous number theorists — go ahead, use the internet and find out more about number theory, history of number theory, the tools and techniques of number theory, the personalities of number theorists, etc. Become a self-learner, self-propeller…if you develop a sharp focus, you can perhaps even learn from MIT OpenCourseWare, Department of Mathematics.
Famous Number Theorists (just my opinion);
5) P. L. Tchebycheff (also written as Chebychev or Chebyshev).
6) Leonhard Euler
7) Christian Goldbach
8) Lejeune Dirichlet
9) Pierre de Fermat
10) Carl Friedrich Gauss
11) R. D. Carmichael
12) Edward Waring
13) John Wilson
14) Joseph Louis Lagrange
16) J. J. Sylvester
11) Leonoardo of Pisa aka Fibonacci.
15) Srinivasa Ramanujan
16) Godfrey H. Hardy
17) Leonard E. Dickson
18) Paul Erdos
19) Sir Andrew Wiles
20) George Polya
21) Sophie Germain
24) Niels Henrik Abel
25) Richard Dedekind
26) David Hilbert
27) Carl Jacobi
28) Leopold Kronecker
29) Marin Mersenne
30) Hermann Minkowski
31) Bernhard Riemann
Perhaps, not-so-famous number theorists (just my opinion):
1) Joseph Bertrand
3) K. Bogart
4) Richard Brualdi
5) V. Chvatal
6) J. Conway
7) R. P. Dilworth
8) Martin Gardner
9) R. Graham
10) M. Hall
11) Krishnaswami Alladi
12) F. Harary
13) P. Hilton
14) A. J. Hoffman
15) V. Klee
16) D. Kleiman
17) Donald Knuth
18) E. Lawler
19) A. Ralston
20) F. Roberts
21) Gian Carlo-Rota
22) Bruce Berndt
23) Richard Stanley
24) Alan Tucker
25) Enrico Bombieri
Happy discoveries lie on this journey…
Question: Prove that any integer can be written as the sum of the cubes of five integers, not necessarily.
We use the identity for , which is an integer for all n. We obtain
Hence, n is equal to the sum
1) Find all prime numbers that divide 50!
2) If p and are both prime numbers, prove that is also prime.
3) (a) If p is a prime, and , prove that in the AP a, , , , , every pth term is divisible by p.
3) (b) From part a, conclude that if b is an odd integer, then every other term in the indicated progression is even.
4) Let denote the nth prime. For , show that .
Hint: Use induction and Bertrand's conjecture.
5) Prove that for every , there exists a prime p with .
Here is a cute example of the power of theory of congruences. Monster numbers can be tamed !!
Find the last two digits of .
A famous mathematician, George Polya said that a good problem solving technique is to solve an analagous less difficult problem.
So, for example, if the problem posed was “find the last two digits of 2479”. How do we go about it? Find the remainder upon division by 100. Now, how does it relate to congruences ? Modulo 100 numbers !
So, the problem reduces to — find out .
Now, what is the stumbling block…the exponent makes the whole problem very ugly. But,
, which means , that is, ,
also, use the fact
So, now we need to compute
1. A simplified form of Fermat’s theorem: If x, y, z, n are natural numbers, and , prove that the relation does not hold.
2. Distribution of numbers: Find ten numbers such that (a) the number is contained in the closed interval (b) the numbers and lie in different halves of the closed interval (c) the numbers , , lie in different thirds of the closed interval (d) the numbers , , and lie in different quarters of the closed interval , etc., and finally, (e) the numbers , , lie in different tenths of the closed interval
3. Is generalization of the above possible?
4. Proportions: Consider numbers A, B, C, p, q, r such that: , , , write the proportion in such a way that in the empty squares, there will appear expressions containing p, q, r only; these expressions being obtained by cyclic permutation of one another expressions.
5. Give an elementary proof of the fact that the positive root of is irrational.
I will give you sufficient time to try these. Later, I will post the solutions.