Quiz at GanitCharcha


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Quiz : On the works of mathematician Srinivasa Ramanujan



1
What is Hardy–Ramanujan number which is the smallest number that can be expressed as the sum of two cubes in two different ways?

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2
Given a positive integer $x$, prime counting function ($f$) outputs number of primes that are less than $x$. Now, the value $f(x) - f(x/2)$ will only change if we obtain another prime which means that $x$ itself is prime. Therefore, $f(x) - f(x/2) \geq n$, if $x \geq R_{n}$ and these $R_{n}$'s are all primes and is referred as Ramanujan Primes. (http://en.wikipedia.org/wiki/Ramanujan_prime)

The smallest Ramanujan Prime number greater than 10 is

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3
Who is the first Indian to be elected as the fellow of the Royal Society?

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4
Who is the first Mathematician to be elected as the fellow of the Royal Society?

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5
The problem of finding the solutions for $a$, $b$, $m$, $n$ and $x$, $y$ such that $a^{3} + b^{3} = m^{3} + n^{3} = x^{3} + y^{3}$ is known as

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6
Ramanujan was awarded Bachelor of Science by Research in 1916 (latter referred as Ph.D) for his work on

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7
A positive integer which has more divisors than any smaller positive integer is called

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8
The first five Highly Composite Numbers are

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9
Which one is not a highly composite number?

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10
Apart from Number Theory and Mathematical Analysis, Ramanujan has great contributions in the area of

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11
Ramanujan worked extensively with the following two English Mathematicians.

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12
A positive integer on a given base if is divisible by the sum of it's digits on the same base, is called a Harshad number.  
Which one is not a Harshad number

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13
In the December 1914 issue of the English magazine 'Strand', a King's college student saw the following puzzle and narrated it to Ramanujan. In a long street there are $n$ number of houses where $n$ is greater than 50 and less than 500. And houses are numbered from left to right as 1, 2, 3, .. so on to $n$. The problem is to find a house with number $x$ such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. Ramanujan while was cooking vegetables in a frying pan gave the most general soloution to the whole class of problems not just the one with the constraint $50 < n < 500.$ Who is the King's college student who narrated the problem to Ramanujan?

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14
In the December 1914 issue of the English magazine 'Strand', a King's college student saw the following puzzle and narrated it to Ramanujan. In a long street there are $n$ number of houses where $n$ is greater than 50 and less than 500. And houses are numbered from left to right as 1, 2, 3, .. so on to $n$. The problem is to find a house with number $x$ such that the sum of the house numbers to the left of it equals the sum of the house numbers to its right. Ramanujan while was cooking vegetables in a frying pan gave the most general soloution to the whole class of problems not just the one with the constraint $50 < n < 500.$ What is the solution for this specific instance of the problem?

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15
Number of positive integers that are less than a given integer $x$ and can be expressed as sum of two square numbers is proportional to $x$ divided by square root of $ln(x)$, i.e., $\frac{x}{\sqrt{ln(x)}}$.
And the constant of proportionality as $x$ grows to infinity is called Landau-Ramanujan constant and its value is

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