University Question Papers CST 292 Number Theory

Number Theory CST 292 Question Paper  July 2021

Number Theory CST 292 Question Paper June 2022

Number Theory CST 292 Question Paper  June 2023

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Number Theory CST 292 KTU IV Semester Honors Course Notes Dr Binu V P 9847390760

About me Syllabus Scheme Teaching Plan Model Question Paper University Question Papers Assignment-1 Module-1  Introduction Well ordering principle Group Ring Fields Group Ring Fields in detail Divisibility Modular Arithmetic GCD- Euclidean Algorithm Bezout's Identity Extended Euclidean Algorithm LCM-Least Common Multiple Linear Diophantine Equations Modular Division Module-II Prime Numbers and Prime-Power Factorisation Fermat and Mersenne Prime Primality Testing and Factorization Miller Rabin Primality Testing Algorithm Fermat  Factorization Congruence Modular Exponentiation Linear Congruences Simultaneous Linear Congruences- Chinese Remainder Theorem Fermat's Little Theorem and Euler's Theorem Wilson's Theorem Module-III Congruence with Prime modulus Congruence with Prime power modulus Pseudoprime and Carmichael Numbers Euler's Totient Function Cryptography The Group of Units - Primitive Roots Module-IV Quadratic Residues-Legendre and Jacobi Symbols Arithmetic Func
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Sum of Squares

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 In this our aim is to determine which integers can be expressed as the sum of given number of squares ;ie; which have the form $x_1^2+x_2^2+\cdots+x_k^2$, where each $x_i \in \mathbb{Z}$, for a given $k$. Definition: For each integer $k \ge 1$, let $S_k=\{n|n=x_1^2+x_2^2+\cdots+x_k^2$ for some $x_1,x_2,\ldots,x_k \in \mathbb{Z} \}$ , the set of all sum of $k$ squares. Example: $S_1=\{0,1,4,9,\ldots\}$ is the set of all squares. By inspection $S_2$, the set of sum of two squares contains 0,1,2,4,5 and 8 but not 3,6, or 7. Lemma: The set $S_2$ contains, the sum of two squares is closed under multiplication ie; if $s,t \in S_2$ , then $st \in S_2$. Proof:Let $s=a_1^2+b_1^2$ and $t=a_2^2+b_2^2$ be the elements of $S_2$, where $a_1,b_1,a_2,b_2 \in \mathbb{Z}$. Then the identity $$(a_1^2+b_1^2)(a_2^2+b_2^2)=(a_1a_2-b_1b_2)^2+(a_1b_2+b_1a_2)^2$$ This shows that $st \in S_2$ Example: We have $8=2^2+2^2$ and $10=3^2+1^2$ so $8.10=80=(2.3-2.1)^2+(2.1+2.3)^2=4^2+8^2$ Note: an equivalent ide
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