Recently, Sir Michael Atiyah claimed to have proved one of the most obscure of mathematical conjectures, the Riemann Hypothesis. Read on to know Read on to know whether one of mathematics’ toughest problems been solved?
Some of the mathematical problems that started as abstract exercise have real-world applications and Riemann Hypothesis (RH) is one of them. Recently, Sir Michael Atiyah, 90-year-old genius, claimed to have proved one of the most obscure of mathematical conjectures, the Riemann Hypothesis.
Atiyah’s ‘solution’ may be wrong, but it has not yet been definitively rejected by peer review. Cryptographers and crypto-currency investors will heave a sigh of relief if the problem remains unsolved since a proof could render their expertise obsolete.
Currently, there is a $1 million Millennium Prize from the Clay Mathematics Institute for finding a valid proof or disproof of Riemann Hypothesis. If Atiyah is wrong, then the prize money is still open for everyone.
In 1859, a German mathematician Bernhard Riemann had an insight into the way how numbers are distributed. But he could not prove it. At that time, he was working with complex numbers i.e. numbers with an imaginary component, ‘i’, which is defined as the square root of minus 1. While calculating a series known as the Zeta Function, he noticed a property that he conjectured to be true for all numbers. If Bernhard Riemann was correct, the Zeta Function tells us a lot about primes numbers that can only be divided by themselves and one.
The RH is true till the 100 billionth test but new theorems have been generated by conditionally assuming Riemann was correct. However, there is an infinity of numbers actually, many infinities and simply because Riemann was correct 100 billion times does not mean the hypothesis is true.
Any proof/disproof could crack the foundations of modern cryptography. It could also lead to insights into quantum theory, since there seems to be a strong correspondence between energy levels in quantum physics and the pure maths of the RH.
Modern digital world, depends on the secure encrypted exchanges, and processing, of digital data. Encryption can be of varying strengths and although encryption may vary in strength, the basic principles are similar for most modern encryption methods. A proof of the RH may include a magic formula that breaks into common encryption systems. For instance, a 4-digit ATM PIN is one of 10,000 combinations that could be guessed in a flash by a modern computer and a bank password of 15 mixed characters is relatively stronger. Military encryption standards are very strong and decoding a military message could, in theory, take very fast computers millions of years.
In modern computers, computation of multiply and divide by simple addition and subtraction. Commonly, encryption is built around the fact that it is far easier to multiply than to divide.
For instance, 101 can be multiplied by 409 which are both primes by simply adding 409 to itself 101 times to reach 41,309.
Dividing 41,309 involves dividing it first by 2, then 3, then 5, 7, and so on, until you discover 101 is a factor.
A computer will subtract 2 from 41,309, all of 20,655 times before rejecting 2 as a factor.
Then it will subtract 3, 13,769 times before rejecting 3 as a factor, and so on.
For instance, ‘ABCD’ could be converted into ‘1234’ (substituting numbers for letters) multiplied by 41,309.
To decode this simple code, you must divide ‘50,975,306’ by 41,309. Try doing this without knowing the factors!
Primes & Encryption
There are many tricks to speed up such computations. But factorization is really hard for large primes. Digital encryption systems are based on using 30-digit, or longer, prime numbers. Crypto-currencies like bitcoin also rely on primes for encryption.
The Greek mathematician Euclid proved that there are an infinity of primes. There is no easy way to find a prime, or to work out how many primes are in a given range. There are 1,229 primes between 1 and 10,000, for example. Proving the RH would, almost certainly, involve developing methods for predicting prime distributions. That would make it much easier to find primes, and break codes.
One of the most critical question is — has Sir Michael Atiyah proved the Riemann Hypothesis?
Sir Michael Atiyah was working on a physics problem of calculating the Fine-Structure Constant which measures electromagnetic attraction between particles, based on his ‘proof’. But Physicists say his Fine Structure Constant calculation is flawed.
He used an approach of reduction ad absurdum by initially assuming the RH was wrong. If it is wrong, certain things must follow. Since those things do not follow, the RH must be correct. This approach is generally treated with suspicion by mathematicians and the initial consensus is, the proof is not convincing. The angle of approach, coming from physics to pure maths, suggests there may be some route to a proof from the real world.
This means that Bitcoin, Ethereum, and other crypto-currencies can trade without fear that somebody can hack the crypto mining process.