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Goldilocks and The Three Proofs of The Infinitude of The Primes

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 A fellow PhD student recently asked me "Have you seen the topological proof that there are infinitely many primes?" and I'm possibly in sparse company being excited by this sentence. It reminded me that I'd seen another insanely overpowered proof that there are infinitely many primes a while ago and so I'd like to present both of these along with Euclid's megaclassic proof. Paul Erdos, the most published mathematician of all time, though agnostic, referred to god as the Supreme Fascist and believed that he held a book of the perfect proof of every theorem in maths. He would exclaim "This one's from The Book" when he saw a particularly elegant proof and the first of our proofs is undoubtedly one of these. Suppose, for contradiction, that there are a finite number of primes. I'm going to consider the resulting number when I multiply all these finitely many primes together and add 1. This new number is not divisible by any of the primes. Indeed...
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Flying with a bike

 This post has absolutely nothing to do with maths but I wasn't really sure where else to put this. I want to share my experience of flying home with a bike after bikepacking as there doesn't seem to be a single comprehensive source of information on this and I relied on anecdotes and Reddit threads for my information. This isn't meant to be a complete guide, just my experience. I booked the flight 2 days in advance once I'd arrived in Prague which alone cost £70. I paid for the largest luggage allowance which was 32kg and cost a further £80. The day before the flight I visited a bike shop to ask if they had a cardboard box for bikes. They didn't but directed me to another bike shop which did. I asked them to hold the box until the next day.  I disassembled the bike outside the bike shop: removed the pedals, seat post, wheels, pannier rack and handlebars. The bike shop lent me a wrench for the pedals. I packed the empty pannier bags into the box and put the rest of ...
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