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Pierre de Fermat gave a criterion for numbers of the form 8a + 1 and 8a + 3 to be sums of a square plus twice another square, but did not provide a proof. [1] N. Beguelin noticed in 1774 [ 2 ] that every positive integer which is neither of the form 8 n + 7, nor of the form 4 n , is the sum of three squares, but did not provide a satisfactory ...
If x is a triangular number, a is an odd square, and b = a − 1 / 8 , then ax + b is also a triangular number. Note that b will always be a triangular number, because 8 T n + 1 = (2 n + 1) 2 , which yields all the odd squares are revealed by multiplying a triangular number by 8 and adding 1, and the process for b given a is an odd ...
The simplified form of Bernoulli's equation can be summarized in the following memorable word equation: [1]: § 3.5 Static pressure + Dynamic pressure = Total pressure. Every point in a steadily flowing fluid, regardless of the fluid speed at that point, has its own unique static pressure p and dynamic pressure q .
There is a straightforward process to convert any linear program into one in standard form, so using this form of linear programs results in no loss of generality. In geometric terms, the feasible region defined by all values of x {\displaystyle \mathbf {x} } such that A x ≤ b {\textstyle A\mathbf {x} \leq \mathbf {b} } and ∀ i , x i ≥ 0 ...
The first perfect squared square discovered, a compound one of side 4205 and order 55. [1] Each number denotes the side length of its square. Squaring the square is the problem of tiling an integral square using only other integral squares. (An integral square is a square whose sides have integer length.)
In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.The dependent variables are replaced by the value of a field at that point in spacetime (,,,) so that the equations of motion are obtained by means of an action principle, written as: =, where the action, , is a functional of the dependent ...
[39] [40] The factorial number system is a mixed radix notation for numbers in which the place values of each digit are factorials. [ 41 ] Factorials are used extensively in probability theory , for instance in the Poisson distribution [ 42 ] and in the probabilities of random permutations . [ 43 ]
The gross world product (GWP), also known as gross world income (GWI), [1] is the combined gross national income (previously, the "gross national product") of all the countries in the world.