| Jens_Kruse_Andersen Prime constellation records. Arithmetic progressions, simultaneous primes, prime gaps. |
| K-Tuple_Permissible_Patterns Findings using an exhaustive search by Thomas J Engelsma. |
| MacTutor_History_of_Mathematics__Prime_Numbers Includes biographies on many mathematicians. |
| MathWorld__Prime_Numbers Index to hundreds of prime-related articles in Eric Weisstein's MathWorld. |
| The_Nine_and_Ten_Primes_Project The discovery of nine and finally ten consecutive primes in arithmetic progression. |
| Notes_and_Literature_on_Prime_Numbers With applets to demonstrate properties of primes. |
| Number_Spiral Explains this method of visually representing the distribution of primes and the relationships between factors and products. |
| On-Line_Encyclopedia_of_Integer_Sequences_(OEIS)__Primes Index to information about many integer sequences involving primes. Site includes numerous other prime-related sequences. |
| Patterns_in_Primes Examples of mostly digit patterns collected by Harvey Heinz. |
| Primality Introductory text on the theory of prime numbers and number fields. Contains proofs of some important theorems including the fundamental theorem of arithmetic. [PDF] |
| Primality_Testing_with_Fermat\'s_Little_Theorem Test numbers for primality and pseudoprimality in Java. |
| Prime_k-tuplets Tony Forbes' extensive collection of special types of prime clusters. |
| The_Prime_Machine Explore interactively the Goldbach conjecture, the distribution of prime twins, the prime number theorem. |
| Prime_Music A musical piece translating prime factorizations to frequencies. Requires RealPlayer. |
| Prime_Numbers Short lesson. |
| Prime_Numbers_and_Factoring Links and references related to primes and factoring. |
| Prime_Numbers_List Browse all prime numbers of less than 10 digits. A prime number checker facility is also included. |
| The_Prime_Page Methods and definitions of finding prime numbers. |
| Radiant_Primes Visualization of prime numbers resembling an astronomical radiant or celestial pathway. |
| Some_Prime_Numbers Sieves and factoring for small numbers. Download primes below 8 billion. |
| Status_of_Search_for_Multifactorial_Primes An organized search for primes of the form (n)(n-k)(n-2k)...+/-1. Includes table of numbers searched and primes found. |
| Thomas_R__Nicely Includes twin prime counts, tables of first occurrence prime gaps, papers. |
| The_Time_Traveler A group with new propositions in various areas of number theory. |
| Visualizing_the_Distribution_of_Prime_Numbers Investigation into patterns in the distribution of the primes by visualizing them. |
| Wikipedia__Prime_Numbers Index to many prime-related articles. |
| World_of_Numbers__World_of_Palindromic_Primes Records, statistics, curios and puzzles about primes reading the same backwards. Compiled by Patrick De Geest. |
| Bipeds_and_Prime_Numbers Graduation address by former professor of philosophy Garrett Barden. Mentions primes as sign of human curiosity. (September 18, 2001) |
| Chair_of_General_Chemistry,_Altai_State_Technical_University Describes scientific work and educational programs, lists group publications. |
| Department_of_Chemistry Contains online course material and lists faculty members. |
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| Fairfield_Industries_Incorporated Fairfield is a geophysical company known for its development and use of advanced technology for acquisition and processing, and its licensing of seismic data. |
| G_A__Ryzhikov_and_M_S__Biryulina,_Consultants Special processing for traditional marine data using "Partial Synergetic Imaging-", or PSI-criterion. |
| Getech Geophysical contractor providing gravity and magnetic services and products to the international oil and mining industry. Offices in Leeds and Houston. |
| Gibson_Consulting Interpretation of gravity & magnetics for hydrocarbon exploration. |
| Input_Output,_Inc_ Provider of land, marine and reservoir seismic instrumentation. Includes comprehensive data sheets. |
| Interactive_Interpretation_&_Training,_Inc_ Outsourcing geoscience for the upstream petroleum industry: geophysics, geology, petroleum engineering, geochemistry, workstation rental. Landmark and GeoQuest software. |
| JGI,_Inc_ Acquisition of seismic and other geophysical data, geothermal surveys and remote sensing. Japan based. |
| Meridian_Surveys Survey and geophysical consultants to the oil and gas industry. |
| Montason_Exploration_Inc_ Geophysical acquisition, processing and interpretation. Electromagnetic technology produces virtual subsurface conductivity log. |
as the variables range over the nonnegative integers. This seems superficially implausiblebecause the expression is a product: the (k + 2) term at the start is multiplied bythe expression in the curly brackets; but in fact all this means is that, when (k + 2)is not prime the second expresion is 0 or negative, and when it is the second expression must be 1.This result was establishedin the paper Diophantine Representation of the Set of Prime Numbers byJames Jones, Diahachiro Sato, Hideo Wada and Douglas Weins in the American Mathematical Monthly, Volume 83, June-July 1976, pages 449 to 464.C.P Willans established a formula to test whether any number, x, is prime. It is
where|a| means the greatest integer not greater than a . The function 
gives the value 1 if x is prime, and 0 if x is composite. Willans then used this to derive a formula for the nth prime number:
These last two results are established in Willan's paper On Formulae forthe Nth Prime Number in the Mathematical Gazette Volume 48, pages413 to 415, 1964. Check out The Prime Pages