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Similarly, the totient is equal to 4 when n is one of the four values 5, 8, 10, and 12, and it is equal to 6 when n is one of the four values 7, 9, 14, and 18. In each case, there is more than one value of n having the same value of φ(n). The conjecture states that this phenomenon of repeated values holds for every n.
The smallest integer m > 1 such that p n # + m is a prime number, where the primorial p n # is the product of the first n prime numbers. A005235 Semiperfect numbers
The central binomial coefficient () is the number of arrangements where there are an equal number of two types of objects. For example, when n = 2 {\displaystyle n=2} , the binomial coefficient ( 2 ⋅ 2 2 ) {\displaystyle {\binom {2\cdot 2}{2}}} is equal to 6, and there are six arrangements of two copies of A and two copies of B : AABB , ABAB ...
(this associates distinct numbers to all finite sets of natural numbers); then comparison of k-combinations can be done by comparing the associated binary numbers. In the example C and C′ correspond to numbers 1001011001 2 = 601 10 and 1010001011 2 = 651 10, which again shows that C comes before C′.
Spoilers ahead! We've warned you. We mean it. Read no further until you really want some clues or you've completely given up and want the answers ASAP. Get ready for all of today's NYT ...
Number of ways to write an even number n as the sum of two primes (sequence A002375 in the OEIS) A very crude version of the heuristic probabilistic argument (for the strong form of the Goldbach conjecture) is as follows. The prime number theorem asserts that an integer m selected at random has roughly a 1 / ln m chance of being prime.
A new study suggests that a diet filled with protein and fiber may be the most effective for weight loss. Here's what a dietitian wants you to know.
Troy Aikman knows the pressures that Caleb Williams faces in his rookie year with the Chicago Bears, and that's why he's concerned about the young quarterback's confidence.