Super Facts about 6-7 - Numberphile

67: A Deep Dive into a Viral Meme & Mathematical Properties

Key Concepts:

• Perfect Number: A positive integer equal to the sum of its proper divisors (excluding the number itself).
• Mersenne Prime: A prime number of the form 2^p - 1, where p is also a prime number.
• Perfect Prime Pair: A pair consisting of a perfect number and a prime number, where the perfect number is generated from a Mersenne prime.
• Sexy Prime: A prime number that is six more or less than another prime number.
• Super Prime: A prime number whose position in the sequence of all prime numbers is also a prime number.
• Super-Super Prime: A prime number whose position in the sequence of all super primes is also a prime number.
• Primeorial: The product of all prime numbers less than or equal to a given prime number.

The 67 Phenomenon: From Meme to Mathematical Reclamation

The discussion begins with the widespread internet meme “67,” which has become a significant distraction in classrooms. The core of the meme involves saying “67” (often accompanied by a specific hand motion) whenever the number six is mentioned, causing disruption. The speaker aims to “reclaim” 67 by exploring its mathematical properties, arguing that numbers are inherently linked to mathematics and should be appreciated for their intrinsic value. A recent incident involving K Star being reprimanded for saying “67” in a classroom further illustrates the meme’s prevalence and the annoyance it causes teachers, like the speaker’s sister-in-law, Cathy.

Perfect Numbers and Mersenne Primes: The Foundation of a Perfect Prime Pair

The speaker introduces the concept of perfect numbers, defining them as numbers equal to the sum of their proper divisors. Examples provided include 6 (1+2+3=6) and 28 (1+2+4+7+14=28). He then explains that perfect numbers are relatively rare. This leads to the discussion of “perfect prime pairs,” which consist of a perfect number and a prime number. The key connection is the Mersenne prime – a prime number of the form 2^p - 1, where p is also prime.

The speaker details several perfect prime pairs:

• 6 & 7: The initial inspiration for reclaiming the number.
• 28 & 29: Another example of a perfect prime pair.
• 33,550,336 & 33,550,337: A larger example, demonstrating the scale of these numbers.

He explains Euclid’s method for generating perfect numbers: if 2^p - 1 is a Mersenne prime, then 2^p-1(2^p - 1) is a perfect number. He notes that all known perfect numbers are even, and the existence of odd perfect numbers remains an unsolved problem in mathematics, with any such number being larger than 10¹⁵⁰⁰. He conjectures that an odd perfect number could not be part of a perfect prime pair.

67 in the Realm of Constants and Sequences

Beyond perfect numbers, the discussion shifts to the appearance of “67” within mathematical constants and sequences.

• Pi (π): The digits 6 and 7 appear consecutively as the 99th and 100th decimal places of pi. They also appear at the 235th and 236th decimal places.
• Euler's Number (e): The digits 6 and 7 appear consecutively as the 60th and 61st decimal places of e.
• Champion's Constant: While 6 and 7 appear in this constant (0.1234567891011…), the speaker dismisses it as a “cheat” because the constant is constructed to include consecutive digits.

Prime Properties of 67: Sexy Primes and Super Primes

The speaker then explores 67’s properties as a prime number itself.

• Prime Number: 67 is a prime number, divisible only by 1 and itself.
• Sexy Prime: 67 is a “sexy prime” because it is part of a sequence of three primes differing by six: 61, 67, and 73 (a “sexy prime triple”).
• Super Prime: 67 is a “super prime” because it is the 19th prime number, and 19 is itself a prime number.
• Super-Super Prime: The discussion extends to “super-super primes” (primes at super prime positions) and the possibility of infinitely many such primes for any finite level of “superness.” He proposes a conjecture that there are no primes at an infinite level of “superness,” suggesting that the numbers involved grow too large. He playfully names this conjecture the “Pila conjecture.”

The 67th Prime Number and a Final Observation

The speaker asks what the 67th prime number is, and it is revealed to be 331. He notes the interesting coincidence that 3+3+1 = 7, linking back to the original number.

Conclusion: Reclaiming 67 Through Mathematical Appreciation

The conversation concludes with a playful attempt to reclaim the number 67 from its meme status by highlighting its rich mathematical properties. The speaker demonstrates that 67 is not merely a disruptive element in classrooms but a number with fascinating connections to prime numbers, perfect numbers, and fundamental mathematical constants. The discussion emphasizes the beauty and complexity hidden within seemingly simple numbers, advocating for a deeper appreciation of mathematics.