So, I was first rather distressed when I found out I was doing Carnival #30; I thought, 30 isn’t that interesting a number, is it? But it turns out that:
1. 30 is the first number that’s a sum of four distinct nonzero squares, that is, . (Sequence)
2. 30 is the first Giuga number. (Sequence) It’s a composite number n such that given any prime factor p, p is a divisor of . In other words:
The factors of 30 are 2, 3, and 5. We need to check all three factors.
2: , and 14 is divisible by 2.
3: , and 9 is divisible by 3.
5: , and 5 is divisible by 5.
3. Speaking of 2, 3, and 5, 30 is the first Sphenic number. That means it has exactly three distinct prime factors. (Sequence)
4. Finally, 30 is the largest very round number: it has the property that all smaller numbers relatively prime to it are prime. (Sequence) This was proved independently by Schatunowsky in 1893 and Wolfskehl (of the Wolfskehl Prize for Fermat’s Last Theorem) in 1901. The proof involves the Bertrand-Chebyshev theorem that given n>1, there exists at least one prime between n and 2n.
So, on to the submissions!
Polymath gives a proof that the nine-point center (aka the Feuerbach circle or the Euler circle) exists. That’s (given a triangle) a circle that passes through: the three midpoints of the sides, the three feet of the altitudes, and the three midpoints of the segments connecting each vertex to the orthocenter.
Alvaro Fernandez at Sharp Brains has a mental visualizing puzzle involving the shape of numbers.
Maria at TCMTechnologyBlog shows off what’s possible with density equalizing maps; the countries of the world inflate and deflate like balloons to model national wealth, or deaths from wars.
The American Institute of Mathematics recently announced the discovery of a “third degree transcendental L-function”. Charles Daney asks the natural question: what on earth are third degree transcendental L-functions?
… and I’m out! Recursivity takes over from here on April 18th for the Carnival #31.
Filed under: Mathematics |