A simple equation that behaves weirdly - Numberphile

Key Concepts

• Diophantine equation: An equation where only integer solutions are sought.
• Quadratic equation: An equation of the form ax² + bx + c = 0.
• Vieta's formulas: Formulas relating the coefficients of a polynomial to sums and products of its roots.
• Vieta involution/jumping: A technique for finding new solutions to Diophantine equations by using Vieta's formulas to "jump" from one solution to another.
• Asymptotic behavior: The behavior of a function as its argument tends towards infinity.
• Growth rate: How quickly a function increases as its argument increases.
• Rational number: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero.
• Blue Skies Research: Research that is driven by curiosity and a desire to expand knowledge, rather than by immediate practical applications.

Solving the Equation x² + y² + z² + w² = xyzw

Initial Problem and a Symmetrical Solution

The video starts with the Diophantine equation x² + y² + z² + w² = xyzw, where x, y, z, and w are positive whole numbers. A symmetrical solution is quickly identified: x = y = z = w = 2. This gives 2² + 2² + 2² + 2² = 2 * 2 * 2 * 2, which simplifies to 16 = 16.

Leveraging the Quadratic Nature of the Equation

The key insight is to treat the equation as a quadratic in one variable (e.g., x) while considering the others (y, z, w) as constants. This allows for the application of the quadratic formula and Vieta's formulas. The equation is rearranged to: x² - (yzw)x + (y² + z² + w²) = 0.

Vieta's Formula and Solution Transformation

The quadratic formula is recalled: For ax² + bx + c = 0, x = (-b ± √(b² - 4ac)) / 2a. Vieta's formula states that the sum of the roots (x and x') of the quadratic equation is -b/a. In this case, x + x' = yzw (since a = 1). This leads to a method for finding new solutions: if (x, y, z, w) is a solution, then (x', y, z, w) is also a solution, where x' = yzw - x.

Generating New Solutions

Starting with the solution (2, 2, 2, 2), the formula x' = yzw - x is applied to generate new solutions. Replacing x with x' = (2 * 2 * 2) - 2 = 6 yields the solution (6, 2, 2, 2). This is verified: 6² + 2² + 2² + 2² = 36 + 4 + 4 + 4 = 48, and 6 * 2 * 2 * 2 = 48.

Iterative Solution Generation and Integer Preservation

The process can be repeated iteratively, transforming any of the variables using the same formula. It's noted that if the initial values are integers, the transformed values will also be integers. Also, if a variable is transformed and then immediately transformed again, it will revert to its original value. To generate new solutions, one should avoid transforming the same variable twice in a row.

Vieta Jumping

The process of using Vieta's formulas to jump from one solution to another is referred to as "Vieta involution" or "Vieta jumping." This technique is related to a problem from the 1986 International Mathematical Olympiad (IMO).

Counting Solutions and the Mystery of Beta

Defining V(R) and the Growth Rate

V(R) is defined as the number of solutions (x, y, z, w) to the equation where x, y, z, and w are positive whole numbers, and x, y, z, w ≤ R (R being an upper bound). The question is how quickly V(R) grows as R increases.

Unexpected Logarithmic Growth

It's stated that V(R) grows like a power of log R, which is unexpected for a Diophantine equation with many variables compared to its degree. Typically, one would expect polynomial growth in R.

The Mysterious Exponent Beta

The growth rate is expressed as V(R) ≈ (log R)^β, where β is an unknown exponent. Empirical calculations suggest that β lies between 2.43 and 2.48.

Asymptotic Behavior and the Constant A

It's known that V(R) is asymptotic to A * (log R)^β as R approaches infinity, where A is a positive constant.

The Open Question: Is Beta Rational?

A question posed by Silverman in 1995 is whether β is a rational number. This is a fundamental question because if β is rational (e.g., 245/100), it might suggest a deeper underlying structure or origin for that specific fraction. If β is irrational, it would indicate an even more complex and mysterious nature to the growth rate of solutions.

Reducing Larger Numbers

The video ends with a brief mention of reducing larger numbers in the equation, hinting at further techniques or simplifications that could be applied.

Conclusion

The video explores a seemingly simple Diophantine equation that leads to surprisingly complex behavior. The use of Vieta jumping allows for the generation of infinitely many solutions. However, the growth rate of the number of solutions below a given bound is governed by a mysterious exponent, beta, whose nature (rational or irrational) remains an open question. This highlights how seemingly elementary mathematical problems can lead to deep and unsolved mysteries.