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For instance, the first counterexample must be odd because f(2n) = n, smaller than 2n; and it must be 3 mod 4 because f 2 (4n + 1) = 3n + 1, smaller than 4n + 1. For each starting value a which is not a counterexample to the Collatz conjecture, there is a k for which such an inequality holds, so checking the Collatz conjecture for one starting ...
Adding 4 hours to 9 o'clock gives 1 o'clock, since 13 is congruent to 1 modulo 12. In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his book Disquisitiones ...
Quadratic Reciprocity (Legendre's statement). If p or q are congruent to 1 modulo 4, then: is solvable if and only if is solvable. If p and q are congruent to 3 modulo 4, then: is solvable if and only if is not solvable. The last is immediately equivalent to the modern form stated in the introduction above.
Let p be an odd prime. The quadratic excess E (p) is the number of quadratic residues on the range (0, p /2) minus the number in the range (p /2, p) (sequence A178153 in the OEIS). For p congruent to 1 mod 4, the excess is zero, since −1 is a quadratic residue and the residues are symmetric under r ↔ p − r.
2-8-8-8-4. Under the Whyte notation for the classification of steam locomotives, a 2-8-8-8-4 has two leading wheels, three sets of eight driving wheels, and four trailing wheels. The equivalent UIC classification is to be refined to (1'D)D (D2') for these engines. Only one 2-8-8-8-4 was ever built, a Mallet -type for the Virginian Railway in ...
4-8-2+2-8-4. Under the Whyte notation for the classification of steam locomotives by wheel arrangement, a 4-8-2+2-8-4 is a Garratt articulated locomotive consisting of a pair of 4-8-2 engine units back to back, with the boiler and cab suspended between them. The 4-8-2 wheel arrangement has four leading wheels on two axles, usually in a leading ...
A lesson plan is a teacher 's detailed description of the course of instruction or "learning trajectory" for a lesson. A daily lesson plan is developed by a teacher to guide class learning. Details will vary depending on the preference of the teacher, subject being covered, and the needs of the students. There may be requirements mandated by ...
[2] [3] [4] They are named after the Italian mathematician Leonardo of Pisa, also known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book Liber Abaci. [ 5 ] Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the Fibonacci ...
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