MATHS

Perfect Numbers 👼

Today we’re going to talk about the idea of a perfect entity, omnipotent (all-powerful) and benevolent (loving). An entity whose nature still evades the understanding of the greatest minds even to this day. I talk, of course, about what the West has come to call…the perfect numbers. Now you might be thinking that I was talking about the idea of God. “Perfect numbers aren’t omnipotent or benevolent!” I hear you cry. This is fair enough; I don’t have any evidence that perfect numbers are benevolent or omnipotent, but it sounded great for the introduction. Anyways, let’s stop with our word games and get to the maths!

What is a perfect number? Simply one that has its factors sum up to its value. 6 is a perfect number: it has factors 1, 2, and 3 (and 1+2+3 = 6). Similarly, 28, 496, and 8128 are part of this group of perfect numbers, and until about 100 AD, it was believed that these were all the perfect numbers that we could know.

It is worth saying that the heavenly words used above (like omnipotent and benevolent), albeit used for the sake of ill-achieved humour in this case, aren’t particularly ill-placed historically! Philo of Alexandria, a Hellenistic philosopher from ~20 CE, in his first-century book On the creation mentions perfect numbers, and claims that the world was created in 6 days and the moon orbits in 28 days because 6 and 28 are perfect.

💡 Things to consider

• Expanding the Search for Perfect Numbers: In the 13th century, Egyptian mathematician Ismail ibn Fallūs made significant strides in expanding the known range of perfect numbers. He discovered three additional perfect numbers: 33,550,336; 8,589,869,056; and 137,438,691,328. Back then, they didn’t have computer programmes like Python, so how in the world did he find these perfect numbers? In fact, he was able to rule out a lot of numbers and make the search much smaller.

• Euclid and Euler: In fact, Fallūs was able to rule out a lot of numbers and make the search much smaller. It was proven by Euclid (often referred to as the ‘father of geometry’) that given a prime p, if 2^{p - 1} is also a prime number, then 2^(p-1)(2^p - 1) is a perfect number. An inverse statement was proven by Leonhard Euler (of course he did) in the 18th century, that every even perfect number has to be of this form. Hence, as we know of 51 Mersenne primes (the largest being 2^82589932 - 1), we know of 51 perfect numbers. The mystery then, is that to this day, in the same way that we do not know whether there are infinitely many Mersenne primes, it is not known if there are infinitely many perfect numbers.

• Some mysteries remain: But wait! What about all the odd perfect numbers, what happened to them? Simple answer, we don’t know. We don’t know if there are any, we’ve never found any, nor have we proved that there can’t be. Weirdly, we know some facts about if there is one. For example, it's bigger than 10¹⁵⁰⁰, and it isn’t divisible by 105 etc. and weird niche facts like this.

🔎 Find out more

• Whilst it is important to know the history of things, part of the joy of maths is in fully understanding why things are true. Look up the proof of the Euclid-Euler theorem online and work through it slowly taking time over the steps. It will prep you for working through proofs in the future!

  • https://penguinmaths.blogspot.com/2019/07/euclid-euler-theorem-proved-visually.html

🍒 The cherry on top

• 🏖️ The Science of Sandcastles: Ever wondered how sandcastles hold their shape despite being made of tiny grains? This PBS article dives into the fascinating physics behind why adding just the right amount of water makes sand stick together. Explore the surprising role of surface tension and capillary action in this classic beach activity. A must-read for anyone curious about Physics or Engineering!

• 🌍 The Carbon Footprint of AI and the Internet: Did you know that the rapid rise of AI and internet usage has a significant environmental cost? This article from Mozilla explores the hidden carbon footprint of our digital lives, from data centers to AI training. Learn how tech innovation impacts the planet and what we can do to make it more sustainable. A vital read for students interested in Environmental Science or Computer Science!

• 🎵 The Politicisation of Music: Is music inherently political? How do you express patriotism through sound? This TED Talk explores the history of music as a political tool. If you have ever wondered about the link between music and politics, this is a great video for you. Have a watch if you are interested in Music, Politics, Or Sociology.

👀 Keep your eyes peeled for…

• 25th November

  • Oxford History Webinar

  • Sheffield German Taster Webinar

• 26th November

  • Oxford History Webinar

  • Oxford Chemistry Workshop

  • Sheffield Electrical Engineering Taster Webinar

  • Sheffield French Taster Webinar

  • UCL English Information Session

• 27th November

  • Oxford History Webinar

  • Sheffield Aerospace Engineering Taster Webinar

  • UEA English Taster Webinar

• 28th November

  • Sheffield Education Taster Webinar

  • Sheffield History Taster Webinar

  • UEA Managing Money at University Webinar

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