An amazing thing about 276 - Numberphile
Numberphile
15 min, 39 sec
A deep dive into aliquot sequences, unsolved problems in number theory, and the Catalan-Dickson conjecture through various number examples.
Summary
- Aliquot sequences involve adding up the proper factors of a number and repeating the process with the resulting sum.
- Numbers explored include deficient, abundant, perfect, amicable, sociable, and aspiring numbers.
- Highlighted the unsolved Catalan-Dickson conjecture, which questions whether all aliquot sequences end in one, a perfect loop, or an amicable loop.
- Unsolved problems include whether five is the only odd untouchable number and the behavior of the Lamer five numbers.
Chapter 1
Starting the conversation with aliquot sequences by picking a number and calculating its proper factors.
- The discussion begins with the choice of the number 276 to explore.
- The concept of proper factors is explained, including how they are summed to proceed in an aliquot sequence.
Chapter 2
Delving into the aliquot process and its linguistic origins.
- The process of continually summing proper factors is named after 'aliquot,' which means 'part of something.'
- Connections are made with other terms like 'quote' and 'quotient,' which also relate to parts or divisions.
Chapter 3
Examining the aliquot sequence of the number 8 and defining deficient numbers.
- Using the number 8 as an example, its aliquot sequence quickly reaches 1, demonstrating a deficient number.
- A deficient number is one where the sum of its proper factors is less than the number itself.
Chapter 4
Exploring the concept of abundant numbers through the aliquot sequence of the number 24.
- When the aliquot sequence results in a number higher than the original, it is called an abundant number, as shown with 24.
- The sequence of abundant numbers can either continue increasing or eventually decline.
Chapter 5
Chapter 6
Introducing the concept of potentially infinite aliquot sequences and the Catalan-Dickson conjecture.
- The question is raised whether there is a number that never reaches 1 or a loop in its aliquot sequence.
- The Catalan-Dickson conjecture posits that all sequences end in one, a perfect loop, or an amicable loop.
Chapter 7
Understanding amicable numbers and the concept of aspiring numbers.
- Amicable numbers are pairs that form loops in their aliquot sequences, like the numbers 220 and 296.
- Aspiring numbers are those that lead to perfect numbers in their aliquot sequences, such as the number 95.
Chapter 8
Discussing Lamer's five numbers and the computational challenges they present.
- Lamer's five numbers are unique and have unsolved aliquot sequences that have not been fully computed.
- The number 276 is highlighted as the lowest of Lamer's five for which we cannot determine the end of its aliquot sequence.
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