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The hardest "What comes next?" (Euler's pentagonal formula)

Mathologer・35 minutes read

Mathologer created a new t-shirt design inspired by Jesko Mathis and Aaron Prince, exploring patterns in connecting dots on a circle to reveal formulas for partition numbers linked to prime numbers. Euler's pentagonal number theorem demonstrates how distinct partitions relate to even and odd numbers, with exceptions for pentagonal numbers, showcasing a visual pattern in partition calculations.

Insights

  • The number of regions created by connecting dots around a circle follows a pattern, with the formula for the nth term being 2 to the power of n, as discussed by Jesko Mathis and Aaron Prince.
  • Euler's pentagonal number theorem showcases exceptions to the rule of equal even and odd distinct partitions, particularly concerning pentagonal numbers, leading to a unique relationship between partition numbers and pentagonal figures.

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Recent questions

  • What is the significance of pentagonal numbers?

    Pentagonal numbers relate to partition number calculations.

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Summary

00:00

"Exploring Patterns in Partition Numbers"

  • Jesko Mathis and Aaron Prince inspired a new Mathologer t-shirt design.
  • The "what's next" question is explored with dots around a circle and connecting them.
  • The number of regions created by connecting dots follows a pattern.
  • The formula for the nth term of the sequence is 2 to the power of n.
  • The mathematicians Bjorn Poonen and Michael Rubinstein found a formula for the number of circle regions in 1997.
  • The partitions of positive integers are examined, revealing patterns.
  • A method involving gaps between blocks is used to determine the number of partitions.
  • The sequence of partition numbers is linked to prime numbers.
  • Leonard Euler investigated partition numbers and discovered patterns.
  • A pattern involving adding and subtracting specific numbers is used to calculate partition numbers.

16:00

"Discovering Patterns in Prime Numbers and Partitions"

  • Challenge to compute the 666th partition number based on discovered connections in mathematics.
  • Explanation of a tweaked partition number machine that reveals a pattern in counter values.
  • The sequence produced by the tweaked machine shows a pattern related to factors of the counter value.
  • Primes have the least number of factors, making the modified machine useful as a prime detector.
  • Example of using the modified machine to determine if numbers like 7, 8, 9, and 10 are prime.
  • Euler's surprise at the connection between prime numbers and partitions, leading to a magic machine.
  • Introduction to pentagonal numbers and their relation to triangular and square numbers.
  • Derivation of the formula for pentagonal numbers and their significance in partition number calculations.
  • Explanation of Ramanujan's complex formula for partition numbers and its incredible accuracy.
  • Visualization of integer partitions using Ferrers diagrams and the hidden relationships within partitions discovered by Euler.

32:16

"Pentagonal Number Theorem: Euler's Partition Patterns"

  • The pentagonal number theorem focuses on distinct partitions of a number with unique summands.
  • Euler proved that the number of even distinct partitions of an integer is equal to the number of odd distinct partitions, except for pentagonal numbers.
  • Exceptions to the rule of equal even and odd distinct partitions occur when the integer is pentagonal.
  • The differences between the number of odd and even distinct partitions align with the pluses and minuses associated with pentagonal numbers.
  • Euler's pentagonal number theorem was visually proven by Fabian Franklin in 1881.
  • Transforming partitions based on diagonal and bottom row lengths results in pairs of even and odd distinct partitions.
  • Exceptions to the pairing rule occur with pentagonal numbers and when the diagonal and bottom rows have a point in common or are of equal length.
  • The number of odd and even distinct partitions for non-pentagonal numbers is equal, while for pentagonal numbers, there is one exception to the pairing.
  • The pentagonal number theorem leads to Euler's recursion formula, connecting partition numbers through specific identities.
  • The recursion formula is demonstrated by splitting dots into groups and transforming partitions to show the relationship between odd and even distinct partitions.

48:10

Euler's machine reveals partition number patterns

  • Euler's machine accurately calculates p(13) by following a pattern of two plus ones followed by two minus ones, demonstrating its correctness not just for p(13) but for all partition numbers.
  • The differences between adjacent partition numbers, illustrated in Ferrers diagrams, showcase a pattern where the orange numbers increase by one and the green odd numbers increase by two, providing a visual explanation for the unique sequence of partition numbers.
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