Number_Theory_and_Diophantine_Approximations

1.

Number Theory and Diophantine
Approximations
• (A Brief Historical and Theoretical Overview)

2.

Origin of Number Theory
• Number theory as a science began with Diophantus.
He studied the existence of integer solutions to equations:
P(x₁, x₂, …, xₙ) = 0
where P is a polynomial with integer coefficients.

3.

Research Direction
• My research is connected with geometric interpretation of Diophantine
approximations.
The theory studies how well real numbers can be approximated by rational
numbers.

4.

Historical Context
• This field has a rich 19th-century history, closely linked with the evolution
of mathematical thought.
Many great mathematicians contributed to its foundation and development.

5.

Dirichlet’s Theorem (1842)
• For any irrational number α and real number Q > 1,
there exist integers p and q such that:
1 ≤ q < Q, |αq - p| ≤ 1/Q
This was the first major result in Diophantine approximation.

6.

Liouville’s Theorem (1844)
• If α is an algebraic number of degree n ≥ 1, and p, q are integers:
|α - p/q| > C/qⁿ
C is a constant depending on α.
This theorem established limits of approximation for algebraic numbers.

7.

Minkowski and Geometry of
Numbers
• Hermann Minkowski — father of the geometry of numbers.
Convex Body Theorem:
Let S be a convex body symmetric about the origin in n-dimensional space.
If Volume(S) > 2ⁿ, then S contains a non-zero lattice point.

8.

Modern Development
• The Department of the Geometry of Numbers was founded at Moscow
State University in 1935.
Founder: L. G. Shnirelman.
The department continues active research on many open problems today.

9.

Conclusion
• From Diophantus to modern times, Diophantine approximation has been a
cornerstone of number theory.
Its blend of algebra, geometry, and analysis continues to shape
mathematical discovery.

10.

References / Acknowledgment
• Based on classical works by Dirichlet, Liouville, and Minkowski.
Historical and theoretical materials from the Faculty of Mechanics and
Mathematics, MSU.