Almost Interesting Facts about 6-7 - Numberphile

67: Fortunate, Lucky, and Almost Integer – A Detailed Analysis

Key Concepts:

• Fortunate Prime: A prime number that satisfies a specific condition related to primorials (products of primes).
• Primorial (nth): The product of all prime numbers less than or equal to the nth prime.
• Lucky Prime: A prime number that survives a specific sieving process (the Sieve of Eratosthenes adapted with changing sieving intervals).
• Lucky Number: A number that survives the sieving process described above.
• Almost Integer: A number that is extremely close to an integer, despite being mathematically irrational.
• Hea Numbers: A specific set of numbers (19, 43, 67, 163, etc.) related to the formation of almost integers involving e and π.
• Modular Function (J-function): A complex function with specific properties used in number theory and string theory.
• OEIS (Online Encyclopedia of Integer Sequences): A database of integer sequences.

I. Fortunate Primes: Definition and Examples

The discussion begins with the number 67 being identified as both a fortunate prime and a lucky prime. A fortunate prime is defined as a prime number that, when added to the product of the first n primes (the primorial), results in the next prime number, with the addition being greater than one.

Example 1: 2 * 3 = 6. The next prime after 6 is 7, requiring an addition of 1. This does not qualify as a fortunate number. However, adding 5 to 6 results in 11, the next prime, making 5 a fortunate number.

Example 2: 2 * 3 * 5 = 30. The next prime after 30 is 37, requiring an addition of 7. Therefore, 7 is a fortunate number.

The speaker notes that 67 is fortunate because adding 67 to the product of the first 11 primes results in the next prime number. A conjecture is presented – though unproven – that all fortunate numbers are prime. It's acknowledged that while fortunate numbers appear to be prime, there's no inherent reason for this to be true based on the definition. The speaker also dismisses the idea that every number is fortunate at some stage.

II. Lucky Primes and the Sieve Algorithm

67 is also identified as a lucky prime, which is a lucky number that is also prime. A lucky number is determined using a modified version of the Sieve of Eratosthenes.

Step-by-Step Process (Sieve Algorithm):

1. List consecutive integers (e.g., 1 to 70).
2. Eliminate every second number (even numbers).
3. Identify the next surviving number (e.g., 3).
4. Eliminate every third surviving number.
5. Repeat steps 3 and 4, using the next surviving number as the interval for elimination, until the process is complete.

The numbers that remain after this process are considered lucky numbers. 67 survives this process, making it a lucky number, and therefore a lucky prime.

III. OEIS Search and Sequence Analysis

The speaker mentions checking the OEIS (Online Encyclopedia of Integer Sequences) for sequences containing 67. While many sequences include 67, no sequence consists solely of 67. One sequence found is a repeating sequence of 67s, which the speaker humorously suggests would be detrimental to students.

IV. 67 as an Almost Integer: A Deep Dive

The discussion shifts to a fascinating property of 67: its involvement in an almost integer. The expression e^π√67 is incredibly close to an integer (14719795274399999...). This is described as an "absurdly close" approximation.

This phenomenon is linked to Hea numbers (19, 43, 67, 163, etc.). The speaker explains that only a small number of these Hea numbers produce almost integers when used in the expression e^π√d. 67 is the second-best Hea number for this purpose.

Technical Explanation (Modular Functions):

The almost integer property is explained through the use of modular functions, specifically the J-function. The speaker states that e^π√d can be expressed as the J-function plus a series of very small terms (scaling with e^-π√d). The J-function yields an integer value only for Hea numbers. This explains why only certain numbers produce almost integers.

The J-function is described as having connections to string theory, highlighting the unexpected interdisciplinary nature of number theory.

V. Hea Numbers and the J-Function

The speaker clarifies that there are only nine known Hea numbers. The closer a Hea number is to the larger values, the closer the resulting expression e^π√d is to an integer. The J-function's integer output for Hea numbers is the underlying reason for the almost integer phenomenon.

Conclusion:

The video explores several intriguing mathematical properties of the number 67, demonstrating its classification as a fortunate prime, a lucky prime, and a key component in the creation of an almost integer. The discussion highlights the interconnectedness of different mathematical concepts – primorials, sieving algorithms, modular functions, and even string theory – and emphasizes the surprising patterns and relationships that can be found within the realm of numbers. The exploration of Hea numbers and the J-function provides a deeper understanding of why 67 exhibits this unique almost-integer property.