Here I tried to prove the Existence Theorem for Laplace Transforms. I don't know what the/a "conventional proof" looks like, but this is what I came up with.
Here's something I just learned: the lucky numbers of Euler.
Euler's "lucky" numbers are positive integers n such that for all integers k with 1 ≤ k < n, the polynomial k² − k + n produces a prime number.
Leonhard Euler published the polynomial k² − k + 41 which produces prime numbers for all integer values of k from 1 to 40.
Only 6 lucky numbers of Euler exist, namely 2, 3, 5, 11, 17 and 41 (sequence A014556 in the OEIS).
The Heegner numbers 7, 11, 19, 43, 67, 163, yield prime generating functions of Euler's form for 2, 3, 5, 11, 17, 41; these latter numbers are called lucky numbers of Euler by F. Le Lionnais.
h/t John Carlos Baez
(@johncarlosbaez) for pointing this out.
🐍 aprxc — A #Python#CLI tool to approximate the number of distinct values in a file/iterable using the Chakraborty/Vinodchandran/Meel’s (‘coin flip’) #algorithm¹.
The fascinating Heegner numbers [1] are so named for the amateur mathematician who proved Gauss' conjecture that the numbers {-1, -2, -3, -7, -11, -19, -43, -67,-163} are the only values of -d for which imaginary quadratic fields Q[√-d] are uniquely factorable into factors of the form a + b√-d (for a, b ∈ ℤ) (i.e., the field "splits" [2]). Today it is known that there are only nine Heegner numbers: -1, -2, -3, -7, -11, -19, -43, -67, and -163 [3].
Interestingly, the number 163 turns up in all kinds of surprising places, including the irrational constant e^{π√163} ≈ 262537412640768743.99999999999925... (≈ 2.6253741264×10^{17}), which is known as the Ramanujan Constant [4].
College Precalculus – Full Course with Python Code by Ed Pratowski and freeCodeCamp focus on the foundation of calculus with Python implementation. This 12 hours course covers the following topics:
✅ Core trigonometry
✅ Matrix operation
✅ Working with complex numbers
✅ Probability
watching #QI. An infinity of mathematicians go into a bar and place an order. "I'll have a pint, a half pint, a quarter pint, an eighth, an ounce.." The barman says "I'll stop you there," and pours out two pints. "The problem with you mathematicians is you don't know your limits" #math#joke
Can someone help me with this #Geometry volume problem?
I'm trying to calculate the volume of water in my storage pond. It is a section of a cone with these dimensions: the bottom is roughly circular and 40 feet diameter; the top is roughly circular and 65 feet diameter. It is 16 feet deep when full.
I probably could've done this in high school geometry class, but that was…er...a very long time ago. #math#FediHelp