Mathematicians were able to discover a pattern for what has long been considered very random: prime numbers. The surprising discovery also suggests that scientists need to be a little cautious when it comes to exploring the perplexing world of primes.
The Mystery Of Prime Numbers
Prime numbers are those divisible only by 1 or by themselves. Primes are the foundation from which all other numbers are constructed. This is because all other numbers are generated by multiplying prime numbers together. Therefore, being able to unravel the mystery of primes is recognized to be a vital aspect in comprehending the basics of arithmetic.
Telling whether a number is prime or not is already determined in advance. In reality, mathematicians do not follow a certain formula or pattern to determine which numbers are prime; hence, making them think that prime number prediction is a random procedure.
Twist Of Events
However, in a surprising twist of events, mathematicians from Stanford University have discovered that the so-called "random" prime numbers are not so random at all.
"It was very weird," says study author Kannan Soundararajan. He compares the experience to looking at a seemingly familiar painting and then realizing that there is a never-before-seen component in it.
All prime numbers, except 2 and 5, all end in 1, 3, 7 or 9. However, while investigating through these numbers, Soundararajan and colleague Robert Lemke Oliver discovered that it is very rare for prime numbers ending in 1 to be followed by another prime number ending in 1.
Such finding should not have surfaced if prime numbers truly follow a random sequence. Consecutive prime numbers are expected to care less about the digits of its neighbors.
More specifically, the team found that for the first hundred million primes, having two consecutive prime numbers ending in 1 happen in just one in 18.5 percent of the time. The scientists explained that if the distribution is random, then consecutive numbers ending in 1 should be present in at least 25 percent of the time.
Prime numbers ending in 3 and 7 have a one in 30 percent rate of occurrence, while a prime number ending in 9 occurs in one in 22 percent of the time.
The authors were able to note this pattern in other number ending combinations, which adds up to the deviations in the expected random pattern.
Randomness is more observed as the number becomes bigger. However, that randomness is quite elusive as the scientists looked at a few trillion, yet the pattern still exists.
While the discovery seems like a perplexing angle to the already complicated puzzle of prime numbers, Soundararajan and Lemke Oliver were able to back it up with an explanation.
Majority of investigations about primes are based on what is called k-tuple conjecture, which was discovered by G. H. Hardy and John Littlewood from the University of Cambridge in the early part of the 20th century.
K-Tuple conjecture is a method used to estimate the frequency of pairs, triples and large groups of primes. This technique is said to be an advanced version of the theory that prime numbers are random.
James Maynard from the University of Oxford, who was not involved in the research, says that mathematicians study primes with the thought that these numbers are 99 percent random. However, the remaining 1 percent should not go unremembered.
Although the k-tuple conjecture has not yet been proven, mathematicians firmly believe that it is accurate. This is because the technique has been useful in anticipating the behavior of prime numbers.
Maynard says the work of Soundararajan and Lemke Oliver may even be a confirmation study of the k-tuple conjecture.
The discovery, as mighty and amusing as it may sound, is yet to be applied in long-term problems involving prime numbers such as the Riemann hypothesis and the twin-prime conjecture. However, the discovery has provided this field of study a little jolt.
Another mathematician named Andrew Granville from Canada's University of Montreal says the discovery provides additional understanding and that every bit of information may help. Even if the things scientists take for granted is indeed inaccurate, it still makes them go back and rethink the things they do know.