Copeland–Erdős constant

Copeland–Erdős constant. The Copeland–Erdős constant is a significant real number in number theory, formed by concatenating the base-10 representations of the prime numbers in ascending order after the decimal point. It was first studied in a 1946 paper by mathematicians Arthur Herbert Copeland and Paul Erdős, who proved it is a normal number in base 10. This constant provides a canonical example of a number constructed from a specific integer sequence that exhibits uniform distribution in its digits, linking the study of prime numbers to the theory of normal numbers.

Definition and basic properties

The constant is formally defined as the decimal expansion 0.23571113171923..., where each subsequent block of digits represents the next prime number. This construction is an example of a concatenation of an infinite sequence of integers. The constant is an irrational number, a fact that follows directly from the infinitude of prime numbers and the uniqueness of its representation. Its irrationality measure is not precisely known, though it is known to be a transcendental number as a consequence of broader results on normal numbers. The constant can also be expressed through an infinite series involving powers of ten, relating it to fundamental concepts in real analysis.

Normality and related results

The landmark result by Arthur Herbert Copeland and Paul Erdős demonstrated that this constant is a normal number in base 10. This means every finite string of digits appears in its expansion with the frequency expected from a uniform distribution. Their proof relied on a general theorem about concatenating sequences of integers that do not grow too quickly, a result now often called the Copeland–Erdős theorem. This work connected deeply to earlier investigations into normal numbers by Émile Borel and the Champernowne constant. The normality result implies the constant is absolutely normal, meaning it is normal in every integer base, a property shared by very few explicitly constructed numbers.

Decimal expansion and approximations

The initial digits of the constant are 0.235711131719232931374143... . The sequence of primes—2, 3, 5, 7, 11, 13, 17, 19, 23, 29—dictates this pattern. High-precision computations of its value have been performed using algorithms for generating prime numbers and efficient concatenation routines. Its continued fraction expansion begins [0; 4, 4, 8, 16, 18, 5, 1, ...], showing no apparent simple pattern, which is typical for such constructed constants. The constant is not a Liouville number, and its approximation by rational numbers is governed by general principles from Diophantine approximation.

Historical context and significance

The constant was introduced in the 1946 paper "Note on Normal Numbers" published in the Bulletin of the American Mathematical Society. This work by Arthur Herbert Copeland and Paul Erdős was part of a mid-20th century flourishing of probabilistic number theory, heavily influenced by the foundational work of Émile Borel. The constant served as a concrete, elementary example that satisfied Borel's normality conjecture for a specific base, advancing the constructive theory of normal numbers. Its creation and analysis exemplify the collaborative and problem-solving spirit of Paul Erdős, whose work with Arthur Herbert Copeland bridged combinatorics and analytic number theory.

Generalizations and related constants

The construction readily generalizes to other bases and sequences. For example, concatenating primes in base 2 or base 16 produces analogous constants, which are also normal in their respective bases by the Copeland–Erdős theorem. Related famous constants include the Champernowne constant (concatenating all positive integers) and the Barbier constant (concatenating the Fibonacci numbers). Variations might concatenate other distinguished sequences like the squares or the Mersenne primes. These constants are often studied in the context of computability theory and algorithmic randomness, linking them to the work of Gregory Chaitin and Andrey Kolmogorov. Category:Mathematical constants Category:Number theory Category:Real numbers