Around the turn of the 20th century, Bolyai Farkás taught mathematics at a university in Transylvania. One day he was too sick to teach, so he sent his mathematically gifted 13-year-old son, János, to teach his classes! As you might imagine, János became quite the mathematician in his own right.
Ninety-five years ago today Bolyai János went to Timișoara, Romania to announce his findings concerning geometry’s fifth postulate. For centuries it was argued that this parallel lines postulate could probably be proved using the previous four of Euclid’s postulates, and it should, therefore, be considered a theorem rather than a postulate. Bolyai János proved that it is indeed something that must be assumed rather than proven, because, by assuming it wasn’t necessary, he was able to create a new and very much non-Euclidean geometry, now known as hyperbolic geometry or Bolyai–Lobachevskian geometry.
Last summer I was walking with some family members through a shopping area behind the opera house in Timișoara, Romania. Suddenly my son, David, excitedly shouted, “Mom, look!” There we stood in front of a street sign marking the strată named for Bolyai János! Here is a picture of me in front of that street sign.
Under his image are several plaques. The first is a replica of part of his proof. Underneath are plaques with a quote from him translated into several languages. Perhaps your favorite language is among them. Here is a close-up of the plaques:
The plaque at the bottom is in English, “From nothing I have created a new and another world. It was with these words that on November 3, 1823, Janos Bolyai announced from Timișoara the discovery of the fundamental formula of the first non-Euclidean geometry.”
We did not get to visit the university named for Bolyai János, but I am thrilled that my son spotted this historic location!
Now I’ll write a little about the number 1277:
- 1277 is a prime number.
- Prime factorization: 1277 is prime.
- The exponent of prime number 1277 is 1. Adding 1 to that exponent we get (1 + 1) = 2. Therefore 1277 has exactly 2 factors.
- Factors of 1277: 1, 1277
- Factor pairs: 1277 = 1 × 1277
- 1277 has no square factors that allow its square root to be simplified. √1277 ≈ 35.73514
How do we know that 1277 is a prime number? If 1277 were not a prime number, then it would be divisible by at least one prime number less than or equal to √1277 ≈ 34.7. Since 1277 cannot be divided evenly by 2, 3, 5, 7, 11, 13, 17, 19, 23, 29 or 31, we know that 1277 is a prime number.
1277 is the sum of two squares:
34² + 11² = 1277
1277 is the hypotenuse of a Pythagorean triple:
748-1035-1277 calculated from 2(34)(11), 34² – 11², 34² + 11²
Here’s another way we know that 1277 is a prime number: Since its last two digits divided by 4 leave a remainder of 1, and 34² + 11² = 1277 with 34 and 11 having no common prime factors, 1277 will be prime unless it is divisible by a prime number Pythagorean triple hypotenuse less than or equal to √1277 ≈ 34.7. Since 1277 is not divisible by 5, 13, 17, or 29, we know that 1277 is a prime number.