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A necessary (but not sufficient) condition for solvability is that n is not divisible by 4 or by a prime of form 4k + 3. [note 3] Thus, for example, x 2 − 3 y 2 = −1 is never solvable, but x 2 − 5 y 2 = −1 may be. [27] The first few numbers n for which x 2 − n y 2 = −1 is solvable are with only one trivial solution: 1
Simplification is the process of replacing a mathematical expression by an equivalent one that is simpler (usually shorter), according to a well-founded ordering. Examples include:
Adam Ondra on the sport climbing route Silence, the hardest free climbing route in the world and the first-ever at 9c (French), 5.15d (American YDS), and XII+ (UIAA).. The two main free climbing grading systems (which include the two main free climbing disciplines of sport climbing and traditional climbing) are the "French numerical system" and the "American YDS system". [2]
Sveriges riksdags health ministry Statens folkhälsoinstitut classified 2C-C as "health hazard" under the act Lagen om förbud mot vissa hälsofarliga varor (translated Act on the Prohibition of Certain Goods Dangerous to Health) as of Mar 1, 2005, in their regulation SFS 2005:26 listed as 2,5-dimetoxi-4-klorfenetylamin (2C-C), making it illegal to sell or possess.
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 ...
The Combat Estimate was introduced by the British Army in 2001, [3] although the military estimate or appreciation process is used widely by militaries around the world. [4] It was developed to simplify and speedup the planning process at Battlegroup (BG) level. [ 5 ]
The notation a < b < c stands for "a < b and b < c", from which, by the transitivity property above, it also follows that a < c. By the above laws, one can add or subtract the same number to all three terms, or multiply or divide all three terms by same nonzero number and reverse all inequalities if that number is negative.
For example, if zeroes are inserted into arbitrary positions of a divergent series, it is possible to arrive at results that are not self-consistent, let alone consistent with other methods. In particular, the step 4c = 0 + 4 + 0 + 8 + ⋯ is not justified by the additive identity law alone. For an extreme example, appending a single zero to ...