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The so-called are of the form +, where and are integers.

6 = 2 3 = (1 + 5)(1 5)

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Small numbers like these are easily factored into primes:

The above replacement of the partial factorization 71 190 by 72 191 is one exampleof what's known, in this context, as a . Michel Marcus has used such substitutions extensively tounearth large numbers with simple abundancies... Conversely, some nontrivial were discovered as a byproduct of that search. The following example (which transforms a number of abundancy 8 into a number of abundancy 15/2 ) was obtained byMarcus on 2009-09-28:

3 and 7 are prime. 24 is 233. 86 is 243. 125 is 53. 214 is 2107.

Thus, there are at least two perfect squares between two (positive)consecutive cubes. For large values of n, there aremany more, of course. Let's see how many: Within k consecutive numbers located aroundsome large integer m, we would expect to find about k / 2m perfect squares. There are k = 3n+3n+1 numbers between m = n and the next cube, so we may expect to find roughly1.5 n perfect squares amongthese. This is, in fact, an excellent estimate sincethe actual count is always one of the two integerswhich bracket that quantity...

The above formula allows you to "easily" compute the LCM of a and b:

Here are some algebraic factorizations, among  :

This allows the GCD of two numbers to be defined as well: Two real numbers are they are proportional to two integers; Their GCD is simply the GCD of those integers multiplied by the common scaling factor.

The GCD of two numbers that are commensurable isbest defined to be . This makes the second fundamental property listed above (as it would entail divisionsby zero). With this convention, the celebrated can be stated compactly. So can the (the irrationality of ).

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Note that u and v are not uniquely defined by , since:

As of January 2018, only 50 Mersenne primes are known, corresponding to the following values of theexponent p. (These are necessarily prime, because if d divides p then 2d-1 divides 2p-1.)

The (n) = -1 (n) = (n) / n of a is 2.
More generally, a number whose abundancy is an is variously called a (MPN) or a . The competing locution "multiply perfect" (used as early as 1907 by R.D. Carmichael) is not recommended ("multiply" would rhyme with "triply", not "apply"). Multiperfect numbers whose abundancy is 2 are called multiperfect numbers.

u p + v q = (u+kq) p + (v-kp) q
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  • x bezout(x,y) + y bezout(y,x) = gcd(x,y) ≥ 0

    +1 and -1 are , they are not .

  • Forsaking that exception,here's how to define ona TI-92, TI-89 or.

    2 + 2 = 2 2

  • Note that is odd for one argumentand even for the other:

    2 = -i (1+i)2 5 = (2+i) (2-i)

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bezout(-x,y) = - bezout(x,y) bezout(x,-y) = bezout(x,y)

The above tables have been arranged to make this particular character appearsin the last row (green shading). Recall that the Kroneker generalization ofthe Legendre symbol obeys the following conditions:

gcd ( 1 , 2 ) = 0gcd ( 1 , (3) ) = 0

Some (real) Dirichlet characters are obtained by generalizingthe (of fame). Generalized versionsof the Legendre symbol often go by other names ( or) which we don't advocate, because suchnomenclature is not technically needed...

Noted (and/or contributors) in alphabetical order:

Except in the trivial cases (i.e., k = 1, 2, 3, 4 or 6) there are indexing scheme with this property (because several automorphisms exist). Thus, the above does not assign unambiguous names to the various Dirichlet characters.

x + y = ( u + v) or (x-u) + (y-v) = 0

If k is 1, 2, 4,the power of an odd prime, or the power ofan odd prime () then the corresponding group is (i.e., it has a ). In that case, we may consider a given primitive root r of themultiplicative group formed by the modulo k, often denoted(/k)*, and state that every character modulo k is obtained by the following definingrelation for (k)-th root of unity z (not necessarily a primitive one).

For the numerical example = 1024, = -15625, = 8404, we obtain:

We have indexed the characters in those tables with the multiplicativeresidues modulo k themselves, in accordance with the aforementioned isomorphism(for the smallest relevant k). This convention makes the above tables symmetrical:

To have smaller constants, we may want to introduce n = -2569 - k

Collectively, the (k) characters modulo k form a with respect to pointwise multiplication ( to the of the residues coprime to k) whose neutral element isthe so-called Dirichlet character modulo k(whose value is 1 for an argument coprime to k and 0 otherwise). It is denoted by the symbol 1 in the above tables.

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