Fermat’s Last Theorem
The Pythagorean theorem says that in a right triangle, a² + b² = c². Now suppose we force the variables to be integers. So the solution a=3, b=4, c=5 is allowed, but a=1.5, b=2, c=2.5 is not allowed, even though it fits the equation. It can be shown that there are an infinite number of solutions with a, b, c all integers.
But what happens if we take this one step up? How many integer solutions are there to a³ + b³ = c³? The answer is none. The same happens with a⁴ + b⁴ = c⁴: no solutions.
a^n + b^n = c^n
In fact, Fermat’s Last Theorem states that for any exponent higher than 2, this equation has no integer solutions. This famous problem, conjectured in 1637, took nearly four centuries to solve, being proved finally by Andrew Wiles in 1995.
The birthday paradox says that if there are 23 people in a room, there is a more than 50% chance that two people have the same birthday. It seems counterintuitive because the probability of having a birthday on any particular day is only 1/365.
But the difference relies on the fact that we only need two people to have the same birthday as each other. If, instead, the game was to get someone with a birthday on a particular day, such as March 14, then with 23 people, there is only a 6.12% chance that someone will have that birthday.
In other words, if there are 23 people in a room, and you choose one person X, and ask, “Does anyone else have the same birthday as X,” the answer will probably be no. But then repeating this on the other 22 people increases the probability every time, resulting in a net probability of more than 50% (50.7% to be more precise).
Division by No 7
If you divide any number by 7, and the answer isn’t an integer, you end up with the sequence 142857 recurring.
1/7 = .142857142857
3/7 = .428571428571
2/7 = .285714285714
6/7 = .857142857142
4/7 = .571428571428
5/7 = .714285714285
International Paper Sizes (e.g. A4) use a 1:√2 ratio. If you cut them in half crosswise, the same ratio will be maintained.
It’s great for scaling up or down.
21st Prime Number
73 is the 21st prime number. Its mirror, 37, is the 12th and its mirror, 21, is the product of multiplying 7 and 3 and in binary 73 is a palindrome, 1001001, which backward is 1001001. 73 is also the best number. Itself and its mirror and 100 plus both numbers are all primes (73, 37, 137, 173).
If you take 73 and 100 more than its mirror, 37 (137) and multiply them together you get 10,001. Which leads to a calculator trick I learned when I was young: Take any four-digit number and multiply it by 73, then multiply the result by 137, your result is the four-digit number repeated twice.
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