Abstract: |
This work is an exploration of certain questions pertaining to the sequence 1, 2, 3, 4, 5,
.... It is divided into three parts. The first part stems from a project conducted in 2007,
headed by K. Ramachandra (Professor at the National Institute of Advanced Studies).
We will give the details of the project here.
Godfrey Harold Hardy remarked, in his lectures on Srinivasa Ramanujan’s life that
Ramanujan, when he was a child in school, discovered that eiθ = cos(θ) + i sin(θ)
(from which the relation eiπ +1 = 0 comes as a consequence). He did this entirely
on his own. The exact age at which he made this discovery is not known, but Hardy
places his age between 7 and 16. In the book, The Music of the Primes, Marcus du
Sautoy says that, after this discovery, Ramanujan “found out a few days later that
Euler had beaten him to this great discovery by some hundred and fifty years. Humbled
and dispirited, Ramanujan hid his calculations in the roof of his house.” Ramanujan’s
original proof and method are unknown. Ramanujan had no access to any material of
modern mathematics save one, Sidney Luxton Loney’s Trigonometry. It is a mystery
how Ramanujan was able to make use of any ideas of complex numbers with the nuances
necessary to reach Euler’s equation. What resulted from this project was an introductory
pamphlet on Trigonometry, where we begin the simple equations (a+b)2 = a2 + 2ab+b2
or (a − b)2 = a2 − 2ab + b2 (one can start from either), and reach Euler’s formula. This
is the probable proof of Ramanujan. This has been published in Mathematics Student.
The second part of the thesis is a simplification of many important problems in the analytic theory of numbers. Here, we provide a discussion of many of the current aspects
of the theory, such as the Riemann zeta function and the Lindel¨of hypothesis. and explain them in elementary terms. We will also discuss a paper of ours published in the Hardy
Ramanujan Journal and relate it to these problems.
The third part, which is new, is a generalization of a problem posed by Paul Erd˝os in
1993 in American Mathematical Monthly. Consider the equation
n! = a!b!
where n>b>a> 1. This equation has an infinite number of solutions. A trivial
solution can be constructed as follows: For any arbitrary natural number a, n = a!, and
b = a! − 1. Nontrivial solutions exist, for example, 10! = 6!7!. Is there a finite number
of non-trivial solutions? In a paper by Florian Luca it was proven conditionally using a
weaker form of the famous “ABC conjecture,” that there is a finite number of nontrivial
solutions: but the “ABC conjecture” – being a relative of the Riemann hypothesis –
may be a long way off from proving. The question reduces to finding a bound on the
distance between n and b. A trivial solution would be of the form n − b = 1. Erd˝os
was able to obtain an upper bound of the difference to 5 log log n. We improve the
absolute constant to 1+�
log 2 and generalize this result, with the same constant, to the
equation n! = a1!a2! ...ak!, for finite k. This result has been accepted for publication in
the Russian journal Matematicheskie Zametki. We proceed with a modification of this
theorem where we obtain comparable bound when we substitute the factorial function
with the product of terms in a class of arithmetic progressions, that is d(2d)(3d)(md) in
the place of m!: The bound we obtain is 1+q/log p
where p is the greatest prime factor of the
common difference in the arithmetic progression. |
Keywords: |
Number Theory, Riemann Zeta Function, Diophantine Equations, Erdos, Hickerson, Factorials, Trigonometry, Elementary Problems, Ramanujan, Hardy, Littlewood, Weyl, Hurwitz Zeta Function |