Mathematicians are always fascinated by assumptions. An excellent mathematical assumption often raises an extremely profound problem in a very precise and concise way, thus inviting people to prove or disprove.
But to make such an assumption is not easy. The assumption itself must be profound enough to pique curiosity and research, but not so obscure that no one can see at all. Many of the most famous problems in the history of mathematics are the assumptions, not solutions, such as Fermat’s last theorem.
Recently, a team of researchers from Technion in Israel and Google in Tel Aviv introduced an automated assumption system they call the Ramanujan Machine, according to the mathematician Srinivasa Ramanujan, who previously developed Thousands of advanced formulas in arithmetic theory without going through any real school. This software system tried to assume many original and important formulas related to the physical constants that often occur in mathematics.
One of the formulas generated by The Machine that could be used to compute the value of a physical constant called a “Catalan number” more efficiently than any ever man-made show before. But the Ramanujan Machine was created not to dominate mathematics, but to support modern mathematicians.
According to the researchers, the entire rules of mathematics can be interpreted into two processes: assumption and proof. The more assumptions that mathematicians make, the more “ingredients” are available to prove and explain problems.
However, it cannot be said that this system lacks ambition. The Ramanujan Machine will “seek to replace the mathematical intuition of the great mathematicians and lead the way in further mathematical research,” the researchers said.
One thing to confirm is that the system is not a versatile math machine. Its main job is to assume formulas to calculate the values of specific numbers called physical constants. The most famous of these is the number pi, the ratio of the circumference to the diameter of a circle. The number pi is called a physical constant because it appears in all mathematical problems, and always maintains a value no matter what the size of the circle is.
The researchers’ system will make assumptions about the values of physical constants (such as pi), written in easy-to-read formulas called inter-fractions. Fractional fractions are basically fractions, but are more complex. The denominator in a conjugate consists of two parts, the second part is a fraction that in itself contains a fraction, and so on until infinity.
Fractional association has long attracted mathematicians for its unusual combination of simplicity and depth: the sum of the value of fractions often equals important constants. In addition to their aesthetic appeal, they are also useful in determining the basic properties of constants.
The Ramanujan machine is built on the foundation of two main algorithms. With great confidence, they discovered that the inter-fraction expressions seem to correspond to physical constants. That confidence is important, because if not, assumptions are easily discarded and have little value.
Each assumption is given in the form of an equation. The idea here is that the numbers on the left side of the equation, a formula that includes the physical constant, will be equivalent to the numbers on the right, a conjugate.
To make these assumptions, the algorithm picks up arbitrary physical constants for the left part and arbitrary conjugates for the right, then computes each side separately to a principal. certain. If the two sides are even, these numbers will be calculated to a higher degree of precision to ensure that their balance is not the result of coincidence due to inaccuracy. Most importantly, we already have formulas that compute the values of physical constants such as pi to an arbitrary degree of precision, so the only barrier to determining whether the two are equal each other is not the calculation time.
Before algorithms like these, mathematicians needed to use their existing mathematical knowledge and theorem to make an assumption. But with automated assumptions, mathematicians can use them to reverse implicit theorems or the more concise results.
But the researchers’ most remarkable finding so far is not hidden knowledge, but a surprisingly important new assumption. This assumption allows the calculation of the Catalan constant, a specialized physical constant whose value is essential to solving many mathematical problems.
The fractions of the newly discovered assumption allowed scientists to compute the Catalan constant at the fastest rate ever, beating previous equations, which took quite a bit of time. computer. It can be said that the Ramanujan Machine marked a new step forward in computing, just like when the computer first defeated the tricks; but this time it was in the making assumptions game.