Oppermann's conjecture is an unsolved problem in
mathematics on the distribution of
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
s.
[.] It is closely related to but stronger than
Legendre's conjecture
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. The conjecture is one of Landau's problems (1912) on prime numbers; , the conjecture has neither be ...
,
Andrica's conjecture, and
Brocard's conjecture
In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (''p'n'')2 and (''p'n''+1)2, where ''p'n'' is the ''n''th prime number, for every ''n'' ≥ 2. The conjecture is named after ...
. It is named after Danish mathematician
Ludvig Oppermann
Ludvig Henrik Ferdinand Oppermann (September 7, 1817 – August 17, 1883) was a Danish mathematician and philologist who formulated Oppermann's conjecture on the distribution of prime number
A prime number (or a prime) is a natural number g ...
, who announced it in an unpublished lecture in March 1877.
Statement
The conjecture states that, for every integer ''x'' > 1, there is at least one prime number between
: ''x''(''x'' − 1) and ''x''
2,
and at least another prime between
: ''x''
2 and ''x''(''x'' + 1).
It can also be phrased equivalently as stating that the
prime-counting function
In mathematics, the prime-counting function is the function counting the number of prime numbers less than or equal to some real number ''x''. It is denoted by (''x'') (unrelated to the number ).
History
Of great interest in number theory is ...
must take unequal values at the endpoints of each range.
[.] That is:
: ''π''(''x''
2 − x) < ''π''(''x''
2) < ''π''(''x''
2 + ''x'') for ''x'' > 1
with ''π''(''x'') being the number of prime numbers less than or equal to ''x''.
The end points of these two ranges are a
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
between two
pronic number A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular number ...
s, with each of the pronic numbers being twice a pair
triangular number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots i ...
. The sum of the pair of triangular numbers is the square.
Consequences
If the conjecture is true, then the
gap size would be on the order of
:
.
This also means there would be at least two primes between ''x''
2 and (''x'' + 1)
2 (one in the range from ''x''
2 to ''x''(''x'' + 1) and the second in the range from ''x''(''x'' + 1) to (''x'' + 1)
2), strengthening
Legendre's conjecture
Legendre's conjecture, proposed by Adrien-Marie Legendre, states that there is a prime number between n^2 and (n+1)^2 for every positive integer n. The conjecture is one of Landau's problems (1912) on prime numbers; , the conjecture has neither be ...
that there is at least one prime in this range. Because there is at least one non-prime between any two odd primes it would also imply
Brocard's conjecture
In number theory, Brocard's conjecture is the conjecture that there are at least four prime numbers between (''p'n'')2 and (''p'n''+1)2, where ''p'n'' is the ''n''th prime number, for every ''n'' ≥ 2. The conjecture is named after ...
that there are at least four primes between the squares of consecutive odd primes.
Additionally, it would imply that the largest possible
gaps between two consecutive prime numbers could be at most proportional to twice the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose ''square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of the numbers, as
Andrica's conjecture states.
The conjecture also implies that at least one prime can be found in every quarter revolution of the
Ulam spiral
The Ulam spiral or prime spiral is a graphical depiction of the set of prime numbers, devised by mathematician Stanisław Ulam in 1963 and popularized in Martin Gardner's ''Mathematical Games'' column in ''Scientific American'' a short time late ...
.
Status
Even for small values of ''x'', the numbers of primes in the ranges given by the conjecture are much larger than 1, providing strong evidence that the conjecture is true. However, Oppermann's conjecture has not been proved
See also
*
Bertrand's postulate
In number theory, Bertrand's postulate is a theorem stating that for any integer n > 3, there always exists at least one prime number p with
:n < p < 2n - 2.
A less restrictive formulation is: for every , there is always ...
*
Firoozbakht's conjecture
*
Prime number theorem
References
{{DEFAULTSORT:Oppermann's Conjecture
Conjectures about prime numbers
Unsolved problems in number theory