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Introduction to SageMath

SageMathCell Suppose you want to expand the following equation $$(x+ \sin x + e^x)^{10} $$ You can use the trinomial theorem and spend lot of time in calculating the coefficients. If you feel that calculating these coefficients is time consuming and the same time can be used for some other purpose, voila! you have an alternative. SageMath is an open source mathematics software which is free to use and quite intuitive. To use SageMath you need not know any programming but if you know it is great. SageMath can be used as an advanced calculator where you can even manipulate algebraic expressions, calculate indefintite integrals, sum of an infinte series and many more. Look at the exxpansion of the above equation. Press the evaluate button <\div>
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Representing primes in binary quadratic forms

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Introduction Prime numbers are of great importance in mathematics. An integer \(p\) is called prime if the only divisors are \(1\) and \(p\). There was always a curiosity about the patterns which can be found in prime numbers. There are sequences of numbers like even numbers, triangular numbers whose \(n^{\text{th}} \) can be easily predicted. For example  \(n^{\text{th}} \) triangular number is given by the formula $$ \frac{n(n+1)}{2}$$ There are also other sequences like Fibonacci numbers whose  \(n^{\text{th}} \) is given by the formula  $$ F_n = F_{n-1} + F_{n-2}$$ with \( F_1=F_0=1\). It turns out that no nice formula for  \(n^{\text{th}} \) prime number is known. So people started looking out for other patterns. In  1640, Fermat wrote a letter to Mersenne in which he mentions " Every prime number which surpasses by one a multiple of four is composed of two squares. Examples are 5, 13, 17, 29, 37, 41, etc. " Formally for any  prime  \( p \) $$ p =x^2 +y^2 \iff p \equiv 1
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