The exhaustive list of topics in Number Theory in which we provide Help with Homework Assignment and Help with Project is as follows:
- Divisibility and Primes:
- Division algorithm, Euclid's algorithm for the greatest common divisor.
- Linear Diophantine equations.
- Prime numbers, fundamental theorem of arithmetic, infinitude of primes.
- Distribution of primes, twin primes, Goldbach conjecture.
- Fermat and Mersenne primes.
- Primality testing and factorization.
Congruences:
- Modular arithmetic.
- Linear congruences.
- Simultaneous linear congruences, Chinese Remainder Theorem.
- An extension of Chinese Remainder Theorem.
Congruences with a Prime-Power Modulus:
- Arithmetic modulo p, Fermat's little theorem, Wilson's theorem.
- Pseudo-primes and Carmichael numbers.
- Solving congruences modulo prime powers.
- Euler's Function and RSA Cryptosystem:
- Definition of Euler function, examples and properties.
- Multiplicative property of Euler's function.
- RSA cryptography.
- Units Modulo an Integer:
- The group of units modulo an integer, primitive roots.
- Existence of primitive roots.
Quadratic Residues and Quadratic Forms:
- Quadratic residues, Legendre symbol, Euler's criterion.
- Gauss lemma, law of quadratic reciprocity.
- Quadratic residues for prime-power moduli and arbitrary moduli.
- Binary quadratic forms, equivalent forms.
- Discriminant, principal forms, positive definite forms, indefinite forms.
- Representation of a number by a form, examples.
- Reduction of positive definite forms, reduced forms.
- Number of proper representations, automorph, class number.
Sum of Powers:
- Sum of two squares, sum of three squares, Waring's problem.
- Sum of four squares.
- Fermat's Last Theorem.
- Continued Fractions and Pell's Equation:
- Finite continued fractions, recurrence relation, Euler's rule.
- Convergents, infinite continued fractions, representation of irrational numbers.
- Periodic continued fractions and quadratic irrationals.
- Solution of Pell's equation by continued fractions.
Arithmetic Functions:
- Definition and examples, multiplicative functions and their properties.
- Perfect numbers, Mobius function and its properties.
- Mobius inversion formula.
- Convolution of arithmetic functions.
- The Riemann Zeta Function and Dirichlet L-Function:
- Historical background for the Riemann Zeta function, Euler product formula, convergence.
- Applications to prime numbers.
- Dirichlet L-functions, Products of two Dirichlet L-functions, Euler product formula.