Dear Math Diary,
Did you know the smallest counterexample to Euler's sum of like powers conjecture is a symmetry which also calculates a particular count of twin primes?
Removing all exponent powers of 5, we get:
27 + 84 + 110 + 133 = 144
LHS = 354, RHS = 144
What does a count of 354 of something have to do with a count of 144 of something?
Well, I'm glad you asked.
The 354th prime number is 2383 (counting primes).
The 144th number that is considered a twin prime is 2383, which is the 354th prime number!
Now, 354 - 144 = 210
Interestingly, 210 is the primorial of 5040.
5040 is 7 factorial = 1 × 2 × 3 × 4 × 5 × 6 × 7
If we only multiply the prime numbers from 1 to 7, we have 2 × 3 × 5 × 7 = 210, which is the primorial of 5040.
5040 is the last known exception from Robin's Theorem. Robin's Theorem is equivalent to the Riemann hypothesis!
We are hot on the trail of this primal music mystery.
We are about to hit the jackpot!
Note: A primorial is the product of all prime numbers less than or equal to a given number. For example, the primorial of 7 (often written as 7#) is 2 × 3 × 5 × 7 = 210.