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, 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.