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For any integer n, n ≡ 1 (mod 2) if and only if 3n + 1 ≡ 4 (mod 6). Equivalently, n − 1 / 3 ≡ 1 (mod 2) if and only if n ≡ 4 (mod 6). Conjecturally, this inverse relation forms a tree except for the 1–2–4 loop (the inverse of the 4–2–1 loop of the unaltered function f defined in the Statement of the problem section of ...
The vertices of the lattice fall into 3 classes numbered 1, 2, and 3, given by the 3 different ways to fill space with hard hexagons. There are 3 local densities ρ 1, ρ 2, ρ 3, corresponding to the 3 classes of sites. When the activity is large the system approximates one of these 3 packings, so the local densities differ, but when the ...
In this example, the ratio of adjacent terms in the blue sequence converges to L=1/2. We choose r = (L+1)/2 = 3/4. Then the blue sequence is dominated by the red sequence r k for all n ≥ 2. The red sequence converges, so the blue sequence does as well. Below is a proof of the validity of the generalized ratio test.
In 1736, Leonhard Euler published a proof of Fermat's little theorem [1] (stated by Fermat without proof), which is the restriction of Euler's theorem to the case where n is a prime number. Subsequently, Euler presented other proofs of the theorem, culminating with his paper of 1763, in which he proved a generalization to the case where n is ...
[2] [4] Oresme's work, and the contemporaneous work of Richard Swineshead on a different series, marked the first appearance of infinite series other than the geometric series in mathematics. [5] However, this achievement fell into obscurity. [6] Additional proofs were published in the 17th century by Pietro Mengoli [2] [7] and by Jacob Bernoulli.
Today's Wordle Answer for #1307 on Thursday, January 16, 2025. Today's Wordle answer on Thursday, January 16, 2025, is FLINT. How'd you do? Up Next:
Wall Street economists expect headline inflation was at 2.9% annually in December, an increase from the 2.7% in November. Prices are set to rise 0.3% on a month-over-month basis, per economist ...
The last step uses the fact that p 2 divides 2 p(p−1) − 1. This follows from Fermat's little theorem, which shows that, for p > 2, 2 p−1 = pk + 1 for some integer k. Raising both sides to the power of p then shows that 2 p(p−1) = p 2 (...) + 1. And now with a similar calculation as above, the following results: