Oppermann's conjecture

In mathematics, Oppermann's conjecture, named after L. Oppermann^[1], relates to the distribution of the prime numbers.^[2] It states that, for any integer x > 1, there is at least one prime between

x(x − 1) and x²,

and at least another prime between

x² and x(x + 1).

Alternative statement

Let π be the prime-counting function:

π(x) = the number of prime numbers less than or equal to x.

Then

π(x² − x) < π(x²) < π(x² + x) for x > 1.

This means that between the square of a number x and the square of the same number plus (or minus) that number, there is a prime number.

If true, this would entail the unproven Legendre conjecture and Andrica conjecture. Oppermann's has not been proved as of December 2010.

References

1. ^ Ludvig Henrik Ferdinand Oppermann (born September 7, 1817; died August 17, 1883), Danish mathematician and philologist (source)
2. ^ Oppermann, L. (1882), "Om vor Kundskab om Primtallenes Mængde mellem givne Grændser", Oversigt over det Kongelige Danske Videnskabernes Selskabs Forhandlinger og dets Medlemmers Arbejder: 169–179, http://books.google.com/books?id=UQgXAAAAYAAJ&pg=PA169&dq=Oversigt+over+det+Kongelige+Danske+Videnskabernes+Selskabs+Forhandlinger+1882+1883+169&hl=en&ei=buD-TLehBML7lweosojECA&sa=X&oi=book_result&ct=result&resnum=1&sqi=2&ved=0CCUQ6AEwAA#v=onepage&q&f=false 

See also

Mathematics portal

• Bertrand's postulate
• Brocard's conjecture
• Prime gap
• Prime number theorem