Model Question Paper CST 292 Number Theory

 

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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 ident
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Quadratic Residues-Legendre and Jacobi Symbols

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Quadratic Congruence   There is a general question whether an integer $a$ has a square root $\pmod{n}$ and if so, how many are there and how one can find them. We know that the solution of the quadratic equation can be found by  $$x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ If we want to apply this to the case where $a,b,c \in Z_n$, then $2a$ to be unit in $\pmod{n}$, so that we can divide by $2a$. We can write the equation in the form $$(2ax+b)^2=b^2-4ac$$ If we can find all the square roots $s$ of $b^2-4ac$ in $\mathbb{Z}_n$, we can then find all the solutions $x \in \mathbb{Z}_n$ of the quadratic equation of the form $2ax+b=s$, or equivalently $x=(s-b)/2a$. If we look at the square roots at $\mathbb{Z}_{15}$, then  1 and 4 have four square roots each($\pm 1 ,\pm 4$ and $\pm 2,\pm 7$ respectively, while the other units have none. Example: Find all the solutions of $x^2-3x+2 \equiv 0 \pmod{15}$ given $a=1, b=−3$ and $c=2,$ we have: $b^2−4ac=(−3)^2−4(1)(2)=9−8=1$ So, $b^2−4ac=1.$ Now, we ne
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