04-05-2024

# Prime numbers, always thought to be random, can actually be predicted

For centuries, mathematicians have been captivated by the elusive and random nature of prime numbers. These numbers, which can only be divided by themselves and the number 1, have long been considered unpredictable.

However, a new study is set to challenge this long-held belief, revealing that prime numbers may not be as random as we once thought.

## Collaborative effort across continents

The brilliant team behind this revolutionary discovery comprises Han-Lin Li, a Visiting Professor in the Department of Computer Science at City University of Hong Kong (CityUHK), Shu-Cherng Fang, the Walter Clark Chair Professor of Industrial and Systems Engineering at North Carolina State University, and Way Kuo, a Senior Fellow at the Hong Kong Institute for Advanced Study, CityU.

“This is a genuinely revolutionary development in prime number theory,” says Way Kuo, who is working on the project alongside researchers from the U.S.

## Predicting random and unpredictable prime numbers

The idea that prime numbers are unpredictable has been a fundamental tenet of mathematics for millennia.

“We have known for millennia that an infinite number of prime numbers, i.e., 2, 3, 5, 7, 11, etc., can be divided by themselves and the number 1 only,” Kuo explains.

“But until now, we have not been able to predict where the next prime will pop up in a sequence of numbers. In fact, mathematicians have generally agreed that prime numbers are like weeds: they seem just to shoot out randomly,” he continued.

However, the research team has developed a method to accurately and swiftly predict when prime numbers will appear.

## Periodic table of primes

The technical aspects of the research may be daunting for most, but the outcome is a user-friendly tool called the periodic table of primes, or the PTP.

This table points to the locations of prime numbers, shedding light on finding future primes, factoring integers, visualizing integers and their factors, identifying locations of twin primes, predicting the total number of primes and twin primes, and estimating the maximum prime gap within an interval, among other applications.

The discovery of the PTP has far-reaching implications, particularly in the realm of cyber security. As Kuo explains, “Primes are already a fundamental part of encryption and cryptography, so this breakthrough means data can be made much more secure if we can predict prime numbers.”

## From color coding to prime number predictions

Interestingly, the breakthrough in prime number research stemmed from the team’s work on systems reliability design and a color coding system that uses prime numbers to enable efficient encoding and more effective color compression.

During their research, the team discovered that their calculations could be used to predict prime numbers.

This serendipitous discovery is a testament to the interconnectedness of various fields of study and the potential for groundbreaking insights to emerge from seemingly unrelated research.

## New era in mathematics

The discovery of the periodic table of primes marks a new era in the study of prime numbers and mathematics as a whole.

By unraveling the mystery of prime numbers, the research team has opened up a world of possibilities for further exploration and practical applications.

As mathematicians and scientists around the world digest this groundbreaking discovery, one thing is certain: the landscape of mathematics will never be the same again.

As discussed above, prime numbers have long been a fascinating subject for mathematicians and number enthusiasts alike.

These unique integers have captivated minds for centuries, and their properties continue to inspire new discoveries and applications.

### What are prime numbers?

A prime number is a positive integer greater than 1 that can only be divided evenly by 1 and itself. In other words, prime numbers have exactly two factors: 1 and the number itself.

For example, 2, 3, 5, 7, and 11 are prime numbers, while 4, 6, 8, and 9 are not.

### Importance of prime numbers

Prime numbers play a crucial role in various fields, including:

• Mathematics: Prime numbers are the building blocks of all integers, as every positive integer can be expressed as a product of prime numbers (the Fundamental Theorem of Arithmetic).
• Cryptography: Many modern encryption systems, such as RSA, rely on the difficulty of factoring large numbers into their prime components to ensure secure communication and data protection.
• Computer Science: Algorithms involving prime numbers are used in hash functions, pseudorandom number generators, and error-correcting codes.

### Interesting facts about prime numbers

• The number 2 is the smallest and only even prime number.
• There are infinitely many prime numbers, as demonstrated by Euclid around 300 BC.
• The distribution of prime numbers among integers appears to be random, but there are some patterns, such as the Twin Prime Conjecture, which states that there are infinitely many pairs of prime numbers that differ by 2 (e.g., 3 and 5, 11 and 13).
• The largest known prime number, as of September 2020, is 2^82,589,933 − 1, which has 24,862,048 digits.

### Current research and future prospects

Mathematicians continue to explore the properties and applications of prime numbers. Some of the most famous unsolved problems in mathematics, such as the Riemann Hypothesis and the Goldbach Conjecture, are related to prime numbers.

As our understanding of prime numbers deepens, we can expect new breakthroughs in fields such as cryptography, computer science, and number theory.

The study of prime numbers continues to inspire and challenge researchers, driving innovation and discovery in mathematics and beyond.

The full study was published in the SSRN Electronic Journal.

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