The mathematician Ken Ribet suggested that only one tenth of 1% of all mathematicians could understand Andrew Wiles's proof of Fermat's Last Theorem (The Times, 24 June, 1993). "And that percentage is probably an overestimate," suggested Ronald Graham.
A prime number can always be found between any interger greater than 1 and its double. This is Bertrand's postulate, also called Bertand-Chebyshev theorem or Chebyshev's theorem, which was first proved by Chebyshev in 1950 using non-elementary methods.
Add the 2nd 4th 6th 8th 10th and 12th digits of any book's 13 digit ISBN. Multiply the sum by 3, then add the rest of the digits. The total will always be divisible 10.
Why? The first 12 digits provide information about the publisher, author and title. The 13th digit is only added as a safeguard against errors. If the calculation above does not result in a total that is divisible by 10, that is an indication that the ISBN has been entered incorrectly.
In the 1995 Homer3 episode of The Simpsons, the equation 178212 + 184112 = 192212 appears in the background. This calculation is false. Only the first 9 digits match. If it were true, it would be a counter-example to Fermat's Last Theorem, which states that xn + yn = zn has no interger solutions for n>2. Fermat's Last Theorem was proved by Andrew Wiles in 1995, 358 years after it was first conjectured by Pierre de Fermat in 1637.
William Shanks, an English amateur mathematician, spent 15 years until 1873 calculating pi to 707 digits by hand. Sadly, only the first 527 digits of his calculation were correct. In 1944, using a mechanical desk calculator, D.F. Ferguson showed that Shanks had miscalculated the 528th digit as 5 instead of 4, which invalidated the subsequent digits of his calculation.
In 2010, Fabrice Bellard, a French programmer, calculated pi to a record 2700 billion decimal digits using a desktop computer bought for less than £2000. The computer took 131 days to complete the calcuation. It would take approximately 49,000 years to recite the number.