1.Riemann
“All things work together for good to them that love God”
Rest here
Georg Friedrich Bernie Hud Riemann
Professor of the University of Göttingen
Born on September 17, 1826, Breslenz
Died on July 20, 1866, Silasga
All things work together
for good to them that love God [1]
Under the tombstone in the courtyard of the church of Ganzolo, the parish of Silasga, is Bernie Riemann. Like all the top mathematicians and physicists, those who touch the edge of truth would be loved by God.
2.Parallel
“I prefer that science and I are two parallel lines. Even if they never intersect, they must not drift away.”
In mathematics, Euclidean Geometry gives five axioms for plane geometry, stated in terms of constructions, of which the fifth axioms is also called Parallel Axiom, because it can be inferred that “In a plane, through a point not on a given straight line, at most one line can be drawn that never meets the given line.”
In the era of F. Gauss (1777–1855), the fifth axiom was questioned. Russian mathematician Nikolay Ivanovitch Lobachevski and the Hungarian mathematician Bolyai stated the fifth public axiom is only a possible choice of the axioms system, not the inevitable geometric truth, that is, thus discovering the non-Euclidean geometry.
In the 19th century, Riemann proposed that “In a plane, through a point not on a given straight line, NO LINE can be drawn that never meets the given line.” This sentence is contrary to our common sense, however Riemann space is a curved space, Riemannian geometry studies a smooth manifold with Riemannian metrics. It pays special attention to angle, arc length and volume; add each tiny part together to get the total quantity.
In 1915, Einstein used the Riemannian geometry and tensor analysis tools to create a new theory of gravity, General Relativity. General Relativity originally originated from Einstein’s realization that gravity is not a force, but a manifestation of geometric bending in space and time. However, there was no mathematical support to explain that gravity is a space-time bending effect. Although Einstein, whose physical intuition is superior to ordinary people, believed that he was right, had been unable to find a mathematical tool to express his thoughts until he learned from friends in mathematics. Riemannian geometry allowed the General Relativity to come out.
3. Riemann Hypothesis
“One of the most beautiful 54 formulas of mankind”
Riemann does have gift in geometry, but Riemannian geometry is too difficult to abstract. The protagonist of today’s article is a hypothesis which is left in his “free time” by 33-year-old Riemann in 1859.
Riemann Hypothesis is that
“The real part of all non-trivial zeros of the Riemann function is 1/2.”
A mathematical expression is used to express that all nontrivial solutions of ζ(n) = 0 are on the line x = 1/2 in the complex plane.
Hard to understand, right? After all, Riemann did not give a proof of this conjecture, and has already tossed the mathematicians for 159 years.
4. Prime, Mathematics, Cryptography
The RSA was proposed in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman. At the time, all three of them worked at the Massachusetts Institute of Technology. The RSA is composed of the first letters of the names of them.
The purpose of this conjecture thrown by Riemann in 1859 was to solve the problem of the law of prime. Once the prime law is solved, almost all Internet encryption methods will no longer be safe, because the main asymmetric encryption, including RSA key encryption, are based on the prime factor decomposition of large numbers.
Looking at the prime table, you will find that the number of primes is getting sparse, and there seems to be no distribution. Mathematicians use prime in cryptography because humans have not yet discovered the laws of prime. If they use the key as encryption, the cracker must try a lot of calculations, that is, using the fastest computer, the process takes too long to lose the meaning of cracking.
The RSA public key encryption algorithm used by major banks and Internet protocols is based on simple prime fact. It is very easy to multiply two large prime, but it is extremely difficult to factorize the product. Therefore, the product can be publicized as an encryption public. key. This asymmetric encryption algorithm for the difficulty of encryption and decryption is called asymmetric encryption.
If Riemann Hypothesis were proved, then the asymmetric encryption algorithm based on large prime decomposition might come to the end, and private key encryption and signature will lose its meaning.
The dangers posed by Riemann Hypothesis are not only affecting banks and the Internet, it may also shake some mathematical foundations. More than one thousand mathematical propositions are premised on the establishment of the Riemann Hypothesis. If the Riemann Hypothesis is proved, all of them can be promoted to theorems; conversely, if the Riemann Hypothesis is falsified, a large part of these mathematical propositions will be buried.
Even some products have used Riemann Hypothesis as a theorem and used it in design. For example, cryptocurrency Primecoin, the difficulty adjustment of finding the prime chains can grow linearly, precisely because the Riemann Hypothesis mentioned that the prime distribution is approximately linear [3].
Sunny King, designer of Primecoin, once said, “I want to use the vast amount of computing power and energy consumed by Bitcoin for scientific research. Finding large prime has always been one of my directions. Because of the irregular distribution of prime, how to adjust the difficulty has bothered me for a long time. Fortunately, I find Riemann Hypothesis.”
Primecoin has modified the Proof of Work consensus of Bitcoin’s hash cash into an algorithm to finding the Canninghum Chain, thereby utilizing the mining energy to the study of prime. The Riemann Hypothesis is used in the difficulty adjustment of the Primecoin. After the algorithm optimization, the block time is maintained at around 60 seconds, which is ten times faster than Bitcoin. The stable operation and mining of the Primecoin in past five years is also a strong testimony to the Riemann Hypothesis.
Some mathematicians pointed out that all the prime numbers in the first kind Canninghum Chain are safe primes[4] and strong primes[5]. The large prime number encryption algorithm is enhanced into a large prime chain encryption algorithm, which may be an enhanced version of RSA algorithm. It is possible to temporarily resist the cryptographic attacks from Riemann Hypothesis is proved to the new encryption algorithm is invented.
Extremely rare, a mathematical conjecture is closely related to so many mathematical propositions. Because of this, the Riemann Hypothesis becomes more and more fascinating.
5. Atiyah
The mathematician Michael Atiyah gave a speech at the 6th Heidelberg Laurel Forum with the theme “Riemann Hypothesis”, and according to his speech summary, he announced the proof.
“Riemann Hypothesis is a well-known unresolved issue. I will report a simple proof using a completely new approach, which is based on von Neumann (1936), Hirzebruch (1954) and The result of Dirac (1918).”
Atiyah is a top mathematician. He won the Fields Medal in 1966 and won the Abel Prize in 2004. However, a large number of mathematicians tried to overcome Riemann Hypothesis and failed. It’s hard to imagine there is a “simple proof” that missed by entire mathematics community, and it was discovered by an 89-year-old mathematician.
People at Reddit pointed out that Atiyah had claimed to solve another mathematical problem without providing proof of details later, and the citations used to prove the Riemann Hypothesis were old and lacking confidence. Although some users praised Atiyah’s courage to challenge the Hypothesis, most people think that he is too old and less likely to discuss and cooperate with young mathematicians, so it is not optimistic that he can handle the large number of detailed mathematical proofs alone.
In this published paper, he mentioned that he wanted to use the cutting-edge knowledge of arithmetic physics and Quantum Computing to focus on the Todd Function, which can be said to provide a solution to the Riemann Hypothesis. However, even if he finds a viable method of proof, he still needs other mathematicians to help clarify the details of the certificate. The academic community also needs a period of review to confirm the proof.
In any case, Atiyah’s report did indeed set off an uproar in the mathematics world. Riemann Hypothesis revitalized study of prime after many years. Let us wait and see, whether the mystery of prime numbers can be penetrated by humans,
Reference
[1] And we know that all things work together for good to them that love God, to them who are the called according to his purpose. (Romans 8:28)
[2] The five axioms of Euclidean geometry are:
- To draw a straight line from any point to any point.
- To produce/extend a finite straight line continuously in a straight line.”
- To describe a circle with any centre and distance/radius.”
- “That all right angles are equal to one another.”
- [The parallel postulate]: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than two right angles.
[3] Prime Coin White Paper
http://primecoin.io/bin/primecoin-paper.pdf
[4]Safe Prime
Safe prime is a type of number that satisfies the 2p+1 form, where p is also a prime number.
https://en.wikipedia.org/wiki/Safe_prime
[5] Strong Prime
https://en.wikipedia.org/wiki/Strong_prime
In cryptography, a prime number p is said to be “strong” if the following conditions are satisfied.
1) p is sufficiently large to be useful in cryptography; typically, this requires p to be too large for plausible computational resources to enable a cryptanalyst to factorise products of p with other strong primes.
2) p − 1 has large prime factors. That is, p = a1q1 + 1 for some integer a1 and large prime q1.
3) q1 − 1 has large prime factors. That is, q1 = a2q2 + 1 for some integer a2 and large prime q2.
4) p + 1 has large prime factors. That is, p = a3q3 − 1 for some integer a3 and large prime q3
