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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.


  • 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.