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  • extraordinary number | planetmath.org
    via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit extraordinary number Define the function G G G for integers n 1 n 1 n 1 by G n σ n n ln ln n assign G n σ n n n G n frac sigma n n ln ln n where σ n σ n sigma n is the sum of the positive divisors of n n n A positive integer N N N is said to be an extraordinary number if it is composite and G N max G N p G a N G N G N p G a N G N geq max G N p G aN for any prime factor p p p of N N N and any multiple a N a N aN of N N N It has been proved in 1 that the Riemann Hypothesis is true iff 4 is the only extraordinary number The proof is based on Gronwall s theorem and Robin s theorem

    Original URL path: http://www.planetmath.org/extraordinarynumber (2016-04-25)
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  • extremal | planetmath.org
    2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11M26 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11A25 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11A25 skos 3AprefLabel 3Flabel FILTER langMatches 28

    Original URL path: http://www.planetmath.org/extremal (2016-04-25)
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  • extremum points of function of several variables | planetmath.org
    Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit extremum points of function of several variables The points where a function of two or more real variables attains its extremum values are found in the set containing the points where all first order partial derivatives vanish the points where one or more of those derivatives does not exist and the points where the function itself is discontinuous Example 1 The function f x y x 2 y 2 1 f x y superscript x 2 superscript y 2 1 f x y x 2 y 2 1 from ℝ 2 superscript ℝ 2 mathbb R 2 to ℝ ℝ mathbb R has a global minimum point 0 0 0 0 0 0 where its partial derivatives f x 2 x f x 2 x frac partial f partial x 2x and f y 2 y f y 2 y frac partial f partial y 2y both equal to zero Example 2 Also the function g x y x 2 y 2 g x y superscript x 2 superscript y 2 g x y sqrt x 2 y 2 from ℝ 2 superscript ℝ 2 mathbb R 2 to ℝ ℝ mathbb R has a global minimum in 0 0 0 0 0 0 where neither of its partial derivatives g x g x frac partial g partial x and g y g y frac partial g partial y exist Example 3 The function f x y z x 2 y 2 z 2 f x y z superscript x 2 superscript y 2 superscript z 2 f x y z x 2 y 2 z 2 from ℝ 3 superscript ℝ 3 mathbb R

    Original URL path: http://www.planetmath.org/extremumpointsoffunctionofseveralvariables (2016-04-25)
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  • factorization of primitive polynomial | planetmath.org
    22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit factorization of primitive polynomial As an application of the parent entry we take the factorization of a primitive polynomial of ℤ x ℤ x mathbb Z x into primitive prime factors We shall see that the procedure may be done by performing a finite number of tests Let a x a n x n a n 1 x n 1 a 0 fragments a fragments normal x normal normal subscript a n superscript x n subscript a n 1 superscript x n 1 normal subscript a 0 a x a n x n a n 1 x n 1 ldots a 0 be a primitive polynomial in ℤ x ℤ x mathbb Z x By the rational root theorem and the factor theorem one finds all first degree prime factors x a x a x a and thus all primitive prime factors of the polynomial a x a x a x If a x a x a x has a primitive quadratic factor then it has also a factor x 2 p x q superscript x 2 p x q displaystyle x 2 px q 1 where p p p and q q q are rationals and conversely For settling the existence of such a factor we treat p p p and q q q as unknowns and perform the long division a x x 2 p x q normal a x superscript x 2 p x q a x x 2 px q It gives finally the remainder b p q x c p q b p q x c p q b p q x c p q where b p q b p q b p q and c p q c p q c p q belong to ℤ p q ℤ p q mathbb Z p q According to the parent entry we bring the system b p q 0 c p q 0 bpq 0 cpq 0 displaystyle begin cases b p q 0 c p q 0 end cases to the form b q 0 c p q 0 bq 0 cpq 0 displaystyle begin cases bar b q 0 bar c p q 0 end cases and then can determine the possible rational solutions p q p q p q of the system via a finite number of tests Hence we find the possible quadratic factors 1 having rational coefficients Such a factor is converted into a primitive one when it is multiplied by the gcd of the denominators of p p p and q q q Determining a possible cubic factor x 3 p x 2 q x r superscript x 3 p superscript x 2 q x r x 3 px 2 qx r with rational coefficients requires examination of a remainder of the form b p q r

    Original URL path: http://www.planetmath.org/factorizationofprimitivepolynomial (2016-04-25)
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  • factors of $n$ and $x^n-1$ | planetmath.org
    query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26C05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26C05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit factors of n n n and x n 1 superscript x n 1 x n 1 Let n n n be a positive integer Then the binomial x n 1 superscript x n 1 x n 1 has as many prime factors with integer coefficients as the integer n n n has positive divisors both numbers thus being τ n τ n tau n Proof If Φ d x subscript normal Φ d x Phi d x generally means the d d d th cyclotomic polynomial Φ d x x ζ 1 x ζ 2 x ζ φ d assign subscript normal Φ d x x subscript ζ 1 x subscript ζ 2 normal x subscript ζ φ d Phi d x x zeta 1 x zeta 2 ldots x zeta varphi d where the ζ j subscript ζ j zeta j s are the primitive d d d th roots of unity then the equation d n d 0 Φ d x x n 1 subscript product d ket n d 0 subscript normal Φ d x superscript x n 1 prod d n d 0 Phi d x x n 1 is true because each n n n th root of unity is also a primitive d d d th root of unity for one and only one positive divisor of n n n The cyclotomic factor polynomials Φ d x subscript normal Φ d x Phi d x have

    Original URL path: http://www.planetmath.org/factorsofnandxn1 (2016-04-25)
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  • factors with minus sign | planetmath.org
    3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11C08 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11C08 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11R18 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11R18 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit The sign cf plus sign opposite number rule

    Original URL path: http://www.planetmath.org/factorswithminussign (2016-04-25)
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  • facts about Riemann--Stieltjes integral | planetmath.org
    3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13A99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13A99 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit facts about Riemann Stieltjes integral If the integrator g g g of the Riemann Stieltjes integral a b f x d g x superscript subscript a b f x d g x int a b f x dg x is the identity function then the integral reduces to the Riemann integral a b f x d x superscript subscript a b f x d x int a b f x dx If the

    Original URL path: http://www.planetmath.org/factsaboutriemannstieltjesintegral (2016-04-25)
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  • Fermat quotient | planetmath.org
    3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A26A42 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Fermat quotient If a a a is an integer not divisible by a positive prime p p p then Fermat s little theorem a k a Fermat s theorem guarantees that the difference a p 1 1 superscript a p 1 1 a p 1 1 is divisible by p p p The integer q p a a p 1 1 p assign subscript q p a superscript a p 1 1 p q p a frac a p 1 1 p is called the Fermat quotient of a a a modulo p p p Compare it with the Wilson quotient w p subscript w p w p which is similarly related to Wilson s theorem Lerch s formula a 1 p 1 q p a w p mod p superscript subscript a 1 p 1 subscript q p a annotated subscript w p pmod p sum a 1 p 1 q p a equiv w p mathop rm mod p for an odd prime p p p connects the Fermat quotients and the Wilson quotient If p p p is a positive prime but not a Wilson prime and w p subscript w p w p is its Wilson quotient then

    Original URL path: http://www.planetmath.org/fermatquotient (2016-04-25)
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