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Colloquium Mathematicum C O L L O Q U I U M .... M A T H E M A T I C U M nnoindentVOL periodCOLLOQUIUM 86 .... 2 0 .... NO periodn 1h f i l l MATHEMATICUM INFINITE FAMILIES OF NONCOTOTIENTS nnoindentBY VOL . 86 n h f i l l 2 0 n h f i l l NO . 1 A period .. F L A M M E N K A M P AND F period .. L U C A .. open parenthesis BIELEFELD closing parenthesisn centerline fINFINITECOLLOQUIUMMATHEMATICUM FAMILIES OF NONCOTOTIENTS g Abstract periodVOL For .any 86 positive integer n let phi open parenthesis 2 0 n closing parenthesis be the NOEuler . 1 function n centerline fBY g of n period A positive INFINITE FAMILIES OF NONCOTOTIENTS integer n is called a noncototient if the equation x minus phi open parenthesis x closing parenthesis = n has n centerline fA. nquad FLAMMENKAMPANDFBY . nquad LUCA nquad ( BIELEFELD ) g no solution x period .. InA.FLAMMENKAMP AND F . L U C A ( BIELEFELD ) this note comma weAbstract give a sufficient . For any condition positive integer on a positiven let φ integer(n) be kthe such Euler that function the geometrical of n: A positive n hspace ∗fn f i l l g Abstract . For any positive integer $ n $ let $ nphi ( n ) $ progression openinteger parenthesisn is called 2 to a the noncototient power of if m the k closing equation parenthesisx − φ(x) = subn has m greater no solution equalx: 1 consistsIn this entirely be the Euler function of $ n . $ A positive of noncototients periodnote , .. we We give then a sufficient use computations condition on to a positive integer k such that the geometrical detect seven suchprogression positive integers(2mk) k≥ period1 consists entirely of noncototients . We then use computations nnoindent integer $ n $ ism called a noncototient if the equation $ x − nphi For any positiveto integer detectn seven let .. such phi positive open parenthesis integers k: n closing parenthesis .. b e the Euler phi function .. of ( x ) = n$ hasnosolution $x .$ nquad In n period .. A For any positive integer n let φ(n) b e the Euler φ function of n: A this note , we give a sufficient condition on a positive integer $ k $ such that the geometrical positive integerpositive n is called integer a noncototientn is called a ifnoncototient the equation xif minusthe equation phi openx − parenthesisφ(x) = n has x closing no parenthesis = n has no solution . For example ; n = 10; 26; 34; 50; 52; 58; 86; 10 are all noncototients . nnoindent progression $ ( 2 ^f m g k ) f m g ngeq 1 $ consists entirely of noncototients . nquad We then use computations to solution periodSierpi For examplen ´ ski and comma Erd hungarumlaut n = 10 comma− 26os comma ( see B 34 36 commain [ 3 50 ] ) comma conj ectured 52 comma that 58 there comma 86 detect seven such positive integers $ k . $ comma 1 0 are allare noncototients infinitely many period noncototients . Sierpi n-acute ski and Erd hungarumlaut-oAn affirmative answer s open t parenthesis o the above see conj B 36 ecture in open was square given bracketby Browkin 3 closing and square n hspace ∗fn f i l l gFor any positive integer $ n $ let nquad $ nphi ( n ) $ nquad b e the Euler bracket closing parenthesisSchinzel conjin [ 1 ectured ] . In thatthe above there arementioned infinitely paper , they showed that 2n · 509203 is a $ nphi $ f u n c t i o n nquad o f $ n . $ nquad A many noncototientsnoncototient period for all positive integers n: Their proof used the number An affirmative509203 answer int o an the essential above conj way ecture . In this was work given , we by extend Browkin the and result of [ 1 ] by giving nnoindent positive integer $ n $ is called a noncototient if the equation $ x Schinzel in opena general square bracketmethod 1 for closing finding square numbers bracketk such period that In 2nk theis abovea noncototient mentioned for paper comma they − nphi ( x ) = n$ hasno showed that 2 to theall powerpositive of integers n timesn: 509203 As a corollary of our method , we have the following : is a noncototient forTheorem all positive . integersLet m n≥ period1 be .. a Their positive proof integer used. the number Then each of the numbers nnoindent509203 in ansolution.Forexample essential way period In this work $, comma n we = extend 10 the , result 26 of open , square 34 bracket , 50 1 closing , square52 bracket, 58 by giving , 86 , 1 0$ areall noncototients. a general method forn finding= 2mk; numbers wherek k2 such f509203 that; 2554843 2 to the; 9203917 power of; 9545351 n k is a; noncototient1645867; 11942443 for ; nnoindentall positiveS integers i e r p i n $ periodnacute ....f Asng a$ corollary ski andof our Erd method $ commahungarumlaut we have− theo following$ s ( :see B 36 in [ 3 ] ) conj ectured that there are infinitely manyTheorem noncototients period65484763 .. Let m .g greater, is a noncototient equal 1 .. be . a positive integer period .. Then each of the numbers n = 2 to the power of m k commaThe where following k in proposition open brace provides 509203 thecomma theoretical 2554843 background comma 920391 for our 7 comma 954535n hspace 1 comma∗fn f i l 1 lTheorem 645867gAn affirmative comma . 1 1 942443 answer comma t o the above conj ecture was given by Browkin and 65484763 closing braceProposition comma is a .noncototientLet k periodbe a positive integer satisfying the following four nnoindentThe followingSchinzel proposition in provides [ 1 ] .the In theoretical the above background mentioned for our paper , they showed that $ 2 ^f n g ncdotTheorem509203 period $ isProposition a noncototient period .. Let for .. k all .. be positive a positive integer integersconditions satisfying $ n the: following . $ nquad four Their proof used the number conditions : line − parenleft i )k is an odd prime . nnoindentline-parenleft509203 i closing in parenthesis an essential k is an way odd prime . In period this work , we extend the result of [ 1 ] by giving ( ii )k is not a Mersenne prime . open parenthesis ii closing parenthesis k is not a Mersenne prime period 2000 Mathematics Subject Classification : 1 1 A 2 5 , 1 1 L 20 , 1 1 L 26 . nnoindent2000 Mathematicsa general Subject method Classification for finding : 1 1 A 2 numbers 5 comma 1 $ 1 L k 20 $ comma such 1 that 1 L 26 period $ 2 ^f n g k $ Work by the second author was supported by the Alexander von Humboldt Founda - isWork a noncototient by the second author for was supported by the Alexander von Humboldt Founda hyphen tion . tion period nnoindentopen squareall bracket positive 37 closing integers square bracket $ n . $ n h f i l l As a corollary of our method , we have the following : [37] n hspace ∗fn f i l l gTheorem . nquad Let $ m ngeq 1 $ nquad be a positive integer . nquad Then each of the numbers n [ n = 2 ^f m g k , where k n in nf 509203 , 2554843 , 920391 7 , 954535 1 , 1 645867 , 1 1 942443 , n ] nnoindent 65484763 ng , is a noncototient . n hspace ∗fn f i l l gThe following proposition provides the theoretical background for our nnoindent Theorem . n hspace ∗fn f i l l g Proposition . nquad Let nquad $ k $ nquad be a positive integer satisfying the following four n begin f a l i g n ∗g conditions : nendf a l i g n ∗g n centerline f $ l i n e −parenleft $ i $ ) k$ is anodd prime . g n centerline f( ii $ ) k $ is not a Mersenne prime . g n centerline f2000 Mathematics Subject Classification : 1 1 A 2 5 , 1 1 L 20 , 1 1 L 26 . g n hspace ∗fn f i l l gWork by the second author was supported by the Alexander von Humboldt Founda − nnoindent t i o n . n [ [ 37 ] n ] 38 .. A period F L AMMENKAMP AND F period L U CA nnoindentopen parenthesis38 nquad iii closingA . F parenthesis L AMMENKAMP .. The AND number F . 2 L to U the CA power of t k minus 1 is composite for all integers t greater equal 1 period n centerlineopen parenthesisf( i ivi i closing ) nquad parenthesisThe number .. The number $ 2 ^ 2f kt is ag noncototientk − period1 $ is composite for all integers $ tThenn thegeq number1 2 to. $theg power of m k is a noncototient for all positive integers m period P r o o f period38 .. AssumeA . F L that AMMENKAMP 2 to the powerAND F .of L mU CA k is a cototient for some m greater equal 1 period Write nnoindentEquation:( open i v parenthesis ) nquad( iii 1The )closing numberThe parenthesis number $ 2t ..k − x k minus1 is $ composite isphi aopen noncototient for parenthesis all integers x closing .t ≥ 1: parenthesis = 2 to theThen power the of m number k period( iv ) $The 2 ^ numberf m g 2kk is $ a noncototientis a noncototient . Then the for number all2 positivemk is a noncototient integers $ m . $Clearly commafor x greater all positive 2 period integers .... Inm: particularP r o o f . comma Assume phi thatopen 2 parenthesismk is a cototient x closing for some parenthesism ≥ 1: i s even periodP r o.... o From f . equationnWritequad openAssume parenthesis that 1 $ closing 2 ^f parenthesism g k $ comma is a it cototient follows for some $m ngeq 1that . $x i s Write even period Write x = 2 to the power of alpha y where y is odd period .. Ifm y = 1 comma we get n begin2 to thef a l power i g n ∗g of m k = x minus phi open parenthesisx − φ x(x closing) = 2 parenthesisk: = 2 to the power of alpha(1) minus 2n tag to the∗f$ power ( of 1 alpha ) minus $g x 1 =− 2 to nthephi power( of alpha x minus ) = 1 comma 2 ^f m g k .
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