Integral Generalized Binomial Coefficients of Multiplicative Functions

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1 Uversty of Puget Soud Soud Ideas Summer Research Summer 015 Itegral Geeralzed Bomal Coeffcets of Multlcatve Fuctos Imauel Che Follow ths ad addtoal works at: htt://souddeas.ugetsoud.edu/summer_research Part of the Number Theory Commos Recommeded Ctato Che, Imauel, "Itegral Geeralzed Bomal Coeffcets of Multlcatve Fuctos" (015. Summer Research. Paer 38. htt://souddeas.ugetsoud.edu/summer_research/38 Ths Artcle s brought to you for free ad oe access by Soud Ideas. It has bee acceted for cluso Summer Research by a authorzed admstrator of Soud Ideas. For more formato, lease cotact souddeas@ugetsoud.edu.

2 Itegral Geeralzed Bomal Coeffcets of Multlcatve Fuctos Imauel Che Mchael Z. Svey Deartmet of Mathematcs ad Comuter Scece Uversty of Puget Soud Setember 10, 015 1

3 The geeralzed bomal coeffcets of some class of fucto f are defed as Q 1 f (. Q Qm m m f 1 f ( 1 f ( The carry sequece of a + b base s deoted as,a,b ad equals 1 f there s a carry the th osto ad 0 f there s ot a carry the th osto. The foudato of ths aer s the followg corollary cotaed Mchael Svey ad Tom Edgar s aer Multlcatve fuctos, geeralzed bomal coeffcets, ad geeralzed Catala umbers [1]. Corollary 8. Let ad m be oegatve tegers. The m s a teger for all multf lcatve fuctos f : N 7 N f ad oly f for all there exsts a s 0 such that, m,m 1 for all < s ad, m,m 0 for all s. I other words, ths corollary states that m s tegral for all multlcatve fuctos f f ad oly f the carry sequece of m lus m s of oe of the three followg forms: , , or Also, to kee thgs cocse,, m,m Ths corollary ca be vsualzed as the followg tragle whch fuctos lke Pascal s tragle; the dot the th row ad mth colum reresets m. A black dot meas that f s tegral whereas a whte dot meas that t s ot tegral. m f Fgure 1: Tragular Reresetato of Corollary 8 Ths aer shows the results of tryg to dscover as much as ossble about Corollary 8.

4 1 Colums Professor Svey ad Professor Edgar already roved whe ( m f the 9th corollary of ther aer. s tegral for m 0, 1, Corollary 9. Let f be a multlcatve fucto ad be a oegatve teger. The ( ( 0 f ad ( are always tegers. 1 f ( ( s a teger f ad oly f (mod 4 or 3 (mod 4. f Observg (, a atter arose where there are alteratg umbers of tegral/otegral 3 f values a row. For examle, as cremets, there would be two tegral values a row, the oe otegral value, the 3 tegral values, ad so o utl these umbers beg to reeat aga. Lemma 1. Let f be a multlcatve fucto ad be a oegatve teger. The ( s 3 f a teger based o ths sequece: , where the frst meas that the frst two values of are tegral values, followed by 1 o-tegral value, followed by 3 tegral values, followed by 3 o-tegral values, ad so o. The sequece the reeats ftely may tmes. Proof. I order to aly Corollary 8, we oly eed to check bases ad 3. (3 for > 3 wll oly have a value the 0th osto whch meas for 1, ɛ, 3,3 1 oly f ɛ, 3, Therefore, there always exsts a s 0 such that ɛ 1 for all < s ad ɛ 0 for all s for all > 3. So to determe whch values of ( 3 suffcet. f, oly checkg ad 3 s Frst cosder. (3 11. Because (3 s two dgts log, we eed oly cosder the 0th ad 1st osto of ( 3 to determe the carres. The frst four values of ( 3 are 00, 01, 10, 11. These values the reeat for the 0th ad 1st osto of ( 3 as we cremet by 4. Out of these four values oly oe that does ot have a s 0 such that ɛ, 3,3 1 for all < s ad ɛ, 3,3 0 for all s s 10. I other words, for (3 + q where q (mod 4, ( s ot a teger. 3 f Next cosder 3. ( Oce aga, we oly eed to cosder the 0th ad 1st osto of ( 3 3. Ths tme there are ow e ossble values that occur ths order: 00 3, 01 3, 0 3, 10 3, 11 3, 1 3, 0 3, 1 3, ad 3. By corollary 8, ( s ot a teger 3 f whe (3 + q where q {6, 7, 8} (mod 9, or whe the 1st two dgts of ( 3 3 are 0, 1, or. The values of that do ot result a tegral value for ( have bee show. Ths 3 f meas that the rest of the values of satsfy corollary 8 for all ad therefore retur 3

5 a tegral value for (. If wrtte out carefully, you wll ed u wth the sequece 3 f Also otce that the lowest commo multle of 4 ad 9 s 36, whch s the sum of every umber ths sequece. From ths roof, I otced a geeral method of determg whe the atter cycles that cossted of oly lookg at the umber of dgts corresodg wth the smaller value betwee m ad m. Lemma. Let d (k be the umber of dgts cotaed the base reresetato of k. I order to determe f ɛ satsfes the codtos of Corollary 8, t s suffcet to oly check the values of ɛ where < d ( or < d (m. Proof. If d ( d (m, the checkg ɛ for < d ( s the same as checkg the etre carry sequece. If d ( < d (m, the for d (, the th osto of ( s 0. Therefore, ɛ 1 oly f ɛ 1 1, or other words, there s a carry a certa osto oly f t was caused by a carry the revous osto. If we cotue to look at d (, ths results a sequece of 0 s ad 1 s that always satsfes the codtos of Corollary 8 ad therefore does ot tell us aythg about whether or ot the etrety of the carry sequece wll satsfy the codtos of Corollary 8. As a result, t s suffcet to oly check the values of ɛ where < d (. Also, because addto s commutatve, t does ot matter whether or ot we check < d ( or < d (m. Theorem 1. For (, where f s a multlcatve fucto ad k s costat, let x k f k 1 f ( s tegral ad x kf k 0 f ( s ot tegral. The the sequece {x k f k} cotas a atter of 1 s ad 0 s that cycles after d(k, where d (k s the umber of dgts cotaed the k base reresetato of k. Proof. I order to aly corollary 8, we have to check ɛ, k,k for all. For values of > k, d (k 1 ad ɛ 1 oly f ɛ 1 1 therefore the carry sequece always satsfes the codto corollary 8. So t suffces to oly check values of k. For values of where d (k > 1, we kow that t s suffcet to oly check ɛ, k,k for < d (k because of lemma. Let {y j (x, } be the sequece such that y j0 j(x, s the frst x dgts of the teger j base. For examle, f 3 ad x, the the corresodg sequece s {00, 01, 0, 10, 11, 1, 0, 1,, 00...}. Ths sequece has the roerty y j (x, y j+ x(x,. Now cosder the sequeces {y k (d (k, } for all k. Based o the revous roerty, each value of wll gve us a cogruece relato: k y k (d (k, (mod d(k k. Combg all cogrueces from each sequece for some ad each k gves us a set of 4

6 cogrueces: k y k (d 1 (k, 1 (mod d 1 (k 1 k y k (d (k, (mod d (k. k y k (d a (k, a (mod da(k a where { 1,,..., a } s the set of all rmes k. As creases, you obta smlar sets of cogrueces. Gve oe of these sets of cogrueces, ( s tegral f ad oly f all values to the k f rght of the equvalet symbol satsfy the codto Corollary 8. Iterrettg tegralty as boolea values, we ca create a sequece {x k } where x k 1 f ( f ( k f s tegral ad x k f k 0 s ot tegral. The Chese Remader Theorem states that each of the good sets of cogrueces have a uque soluto mod good sets of cogrueces reeat every k d(k values of. a 1 d 1 (k k d(k []. Therefore, these It s ukow whether the cycle gve by ths theorem s the shortest ossble cycle. To demostrate ths theorem, cosder (. The base reresetatos are f , so d (7 3, d 3 (7, d 5 (7, ad d 7 (7. The atter ( the 7 f cycles every d( teratos of. 7 Cetral Bomal Coeffcets The cetral bomal coeffcets are defed as ( for 0, or equvaletly, the ots f the drect ceter of the bomal coeffcet tragle. I order to aly Corollary 8, the carry sequece of added to tself, ɛ,,, must be checked. If, the the th osto of has to be 1 order for a carry to occur. Ths meas that the base reresetato of must cosst of all 1 s order to satsfy Corollary 8 s codto. Lemma 3. For a teger, there exsts some s 0 such that ɛ,, e,, 0 for all s f ad oly f x 1 for some x 0. 1 for all < s ad Proof. Whe addg a umber to tself base, a carry occurs at some osto f ad oly f 1. Ths also meas that f 0, there wll be o carry at osto. Therefore, the oly way for there to exst a s such that ɛ,, 1 for all < s ad e,, 0 for all s s for the umber to cosst of all 1 s or 0 s. But of course, a umber cosstg of all 5

7 0 s s smly 0 whch s trval. So a umber base must ether be 0 or cosst of all 1 s to satsfy ths codto, ad base 10, ths s equvalet to x 1 for x 0. Lemma 3 restrcted the values of to the form x 1. Lookg at the base three reresetato of ths form, t was observed to ever have a the 0th osto. Lemma 4. For x 1, mod 3 s ether 0 or 1. Proof. Whe x 1, x 1 1 (mod 3. Wheever x 1 1 (mod 3, x+1 1 x+1 + ( + 1 (mod 3 ( x (mod 3 ( x (mod 3, x 1 1 (mod 3 3 (mod 3 0 (mod 3. Now suose x 1 0 (mod 3, x+1 1 x+1 + ( + 1 (mod 3 ( x (mod 3 ( x (mod 3, x 1 0 (mod 3 1 (mod 3. Therefore, x 1 (mod 3 always results a 0 or 1. Lemma 4 tells us that the carry sequece must cosst of the form order to satsfy Corollary 8 s codto for 3. Therefore, t s suffcet to kow f the base 3 reresetato of x 1 has a t sce ɛ 3,, 1 oly f the th osto of equals. Ths roblem, however, s show to be a credbly dffcult roblem to solve by a uaswered cojecture by Erdos ad Graham whch states that the terary reresetato of x has at least oe for x > 8 [3]. Lemma 4 shows that the terary reresetato of x 1 has ether a 0 or 1 the 0th osto, whch meas that the terary reresetato of x has ether a 1 or the 0th osto. Sce the terary reresetatos of x 1 ad x oly dffer the 0th osto, the f we could show that the terary reresetato of x 1 cotas at least oe for x > 8 t would mly the Erdos ad Graham cojecture. So rather tha solvg a roblem that s eve more dffcult tha oe that has bee usolved by great mathematcas, teratg through the frst 1,000,000 values of x seemed reasoable. 6

8 Theorem. For 1,000,000 1, the cetral bomal coeffcets (, f Z+, are tegral for all multlcatve fuctos f oly for 0, 1, 3. Proof. By Corollary 9, we kow that for 0, 1, ( s tegral. By lemma 3, we kow that f x 1 for x Z +, the ( s ot tegral. So ow let x 1 for x Z +, whch satsfes Corollary 8 for. Let us deote to be the value of the dgt at the th osto of base 3. Oly whe does ɛ 3,, 1 ad by lemma 4, we kow that 0 s ever. Ths meas that f for > 0, the Corollary 8 s ot satsfed ad ( s ot tegral. After teratg f through 1,000,000 values of x, the oly values of x whch Corollary 8 s satsfed for 3 are x 0, 1,, 5, 8. The corresodg values of are 0, 1, 3, 31, ad 55. We ve already show that ( for 0, 1. f For 3, we have already checked bases ad 3 so we eed oly check base 5. Let ɛ,a,b be the etre carry sequece of a + b base. So ɛ 5,3,3 1 ad therefore satsfes Corollary 8 s codto. Therefore, ( 6 s tegral by Corollary 8. 3 f For , ɛ 7,31,31 10 ad does ot satsfy Corollary 8 s codto. Smlarly, for , ɛ 7,55, ad does ot satsfy Corollary 8 s codto. Therefore, by Corollary 8, ( s ot tegral for 31 or 55. f A robablstc argumet ca also be made that for > 1,000,000 base 3, would have more tha dgts so the chace of the exstece of a somewhere s extremely hgh. Assumg there at each osto there s a equal robablty for a 0, 1, or to occur ad that these robabltes are cosstet through all ostos, the robablty that there s at least oe s aroxmately 100%. Of course ths s ot a exact roof but shows that theorem more tha lkely ales to all N. I a effort to geeralze the cetral bomal coeffcets to ( x where x N, a geeralzato for ( f, where s rme, was foud stead. f Theorem 3. The geeralzed bomal coeffcets ( where s rme, N f 0 (atural umbers cludg 0, ad f s a multlcatve fucto are ot tegral whe s a multle of. Proof. I order to aly Corollary 8, we eed to look at the carry sequece ɛ,, ɛ,( 1,. The base reresetato of x j for j N 0 ad x N, or multles of owers of, wll always have a 0 the 0th osto. Therefore, f there s a carry osto > 0, the ( s ot tegral f f x j. a We ca use the formula mod b to determe the cth dgt of a umber a base b c b. Let x be the frst dgt of x j base to have a ozero value. Usg the above formula, the value of the xth dgt of base s 7

9 x j mod j mod x Smlarly, the value the xth osto of ( 1 ( 1 x j base s ( 1 x j x mod (j j mod j mod j mod j mod Addg the xth osto of ad ( 1 base the yelds the result x 1 j mod x 1 j mod 0 mod. Ad sce the xth osto s a ozero value, j mod s also ozero. Ths meas that there s always a carry the xth osto. Therefore, by Corollary 8, ( s ot tegral whe s a multle of. f 3 Multlcatve Fuctos That Are Not Dvsble The ma theorem of Mchael Svey ad Tom Edgar s aer s defed as follows: ES Theorem 1. Let f : N N be a multlcatve fucto. The ( m f ( f( ɛ 1 ɛ 1 ( 0 ( f( +1 ɛ. f( Usg ths theorem, Mchael Svey ad Tom Edgar roved that multlcatve fuctos that are dvsble always have tegral geeralzed bomal coeffcets. ES Theorem. Let f : N N be a multlcatve fucto. The f satsfes f( r f( r+1 for all rmes ad oegatve tegers r ff f s dvsble. No-dvsble multlcatve fuctos f, however, are ot always tegral for ay ( but m f have very terestg atters. 3.1 Ruler Fucto The Ruler Fucto r( s a multlcatve, odvsble fucto defed as the largest ower of that dvdes (OEIS A [4]. For examle, r(4 3 because the largest ower of that dvdes 8 s 3. The multlcatve defto of the Ruler Fucto s 8

10 e + 1, r( e 1, > The same kd of tragular reresetato used for Corollary 8 ca be used for secfc fuctos. The Ruler Fucto s tragular reresetato s show the followg fgure. Fgure : Tragular Reresetato of the Ruler Fucto If we restrct the carry sequece ɛ to the form that satsfes Corollary 8, or ɛ 1 for all 0 k ad ɛ 0 for > k, usg ES Theorem 1 we obta ( m f 0 ( ( r( +1 ɛ r( 0 ɛ ( ( > k + k + 1 k +, 0 ( ɛ 1 1 9

11 whch further demostrates that the Ruler Fucto s tegral wheever Corollary 8 s codto s satsfed. Showg whe the Ruler Fucto s eve ad whe t s odd wll be useful for rovg whe ( s tegral because of the ature of the geeralzed bomal coeffcets formula. m r Theorem 4. The Ruler Fucto r( s eve whe s of the form j+1 (1 + ad odd whe s of the form 4 j (1 + for j 0 ad 0. 0} {4 j (1 + j 0, 0} N. Also, { j+1 (1 + j 0, Proof. Cosder the form x + x+1 for x 0 ad 0. Ths form s the owers of multled by every odd umber, x + x+1 x (1 +, whch reresets every atural umber accordg to the Fudametal Theorem of Arthmetc. Therefore, { x + x+1 x 0, 0} N. We ca slt ths form to two dsjot sets: x + x+1 wth eve x ad x + x+1 wth odd x deoted as A ad B resectvely. Frstly, cosder r( where A: x+1 + x+ ad the largest ower of that dvdes s x+1. So r(a x + 1. But sce x s eve ths set, x + 1 must be odd. Therefore, r( s odd for A. Aother way to wrte A s x + x+1 j + j+1, eve x j for j 0 4 j + 4 j 4 j (1 + Now cosder r( where B. The same exact math occurs but sce x s odd ths set, x + 1 must be eve. Therefore, r( s eve for B. Aother way to wrte B s x + x+1 j+1 + j+, odd x j + 1 for j 0 j+1 (1 +. We kow that the frst two colums of the Ruler Fucto are tegral because of Corollary 9. Usg theorem 4, we ca rove geeralzatos about the ext coule colums of the Ruler Fucto. Theorem 5. Let r be the Ruler Fucto. The geeralzed bomal coeffcets ( ( ad r +1 s tegral f ad oly f s of the form r j+1 (1 + for j 0 ad 0. 10

12 Proof. Pluggg the Ruler Fucto to the Geeralzed Bomal Coeffcet formula, we obta ( r 1 r( 1 r( 1 r( r(r( 1 r(1r( r(r( 1. Ths meas that ( s tegral whe r(r( 1 s eve. For odd values of, r( 1, r whch meas oe of the terms r( or r( 1 wll equal 1 ay gve case. Therefore, fdg a value of where r( s eve meas that ( the follows from theorem 4. r ad ( +1 r are tegral. The result Theorem 6. The thrd colum of the geeralzed bomal coeffcets of the Ruler Fucto, ( 3, are ot tegral whe s of the form r 4j ( for j 1 ad 0. Proof. Reducg ( 3 r, we obta ( 3 r 1 r( 3 1 r( 3 1 r( r(r( 1r( r(1r(r(3 r(r( 1r(. Therefore, ( s ot tegral whe r(r( 1r( s odd, whch wll oly be odd f 3 r each of the three terms are odd. By theorem 4, we kow that r( s odd whe s of the form 4 j (1 + for j 0 ad 0, so we smly eed to fd whe ths form has three cosecutve tegers. Whe j 0, we have all the odd umbers. Ths meas that f we fd whe 4 j (1 + s eve, we have foud three cosecutve tegers wth ths form. I other words, f s odd, r( r( 1 so we eed oly fd whe 1 s eve ad of the form 4 j (1 +. Ths s smly whe j 1. So whe 1 s of the form 4 j (1 + for j 1 ad 0, the ( s tegral. 3 r The followg theorem geeralzes the ars of tegral rows that ca be observed the tragular reresetato. Theorem 7. The geeralzed bomal coeffcets of the Ruler Fucto ( m r whe k 1 ad k for k > are tegral

13 Proof. Whe k 1 for k > 0, ( m r 1 r( m 1 r( m 1 r( r(k 1r( k...r( k m. r(mr(m 1...r(r(1 Notce that r( k m r(m whe 0 m k 1 because the hghest ower of that dvdes k+1 m wll be the ower of that dvdes m. Therefore, every term the umerator has a corresodg term the deomator that cacels t. So, ( 1 whe m r k 1. Whe k for k > 0, ( m r r(k r( k 3...r( k mr( k m 1. r(mr(m 1...r(r(1 Comared wth the k 1 case, the dfferece here s that the terms r( k m 1 ad r(1 do ot have a corresodg term to cacel wth sce there s o r(m + 1 term the deomator ad o r( k 1 term the umerator. However, ths leaves us wth ( r(k m 1 r( k m 1, so ( mr m r(1 s also always tegral for r k. 3. Dvsor Fucto The Dvsor Fucto τ( s the umber of dvsors of cludg tself (OEIS A [5] ad has the followg roertes: 1. Multlcatve wth τ( e e + 1. τ( s odd f ad oly f s a erfect square 3. τ( for rme. The corresodg tragular reresetato of τ s show fg. 3. Cosderg aga f ɛ 1 for all 0 k ad ɛ we obta ( m f ( 0 ( ɛ for > k, usg ES Theorem 1 ( k + k + 1 (k +. Notce that ths result s that exact same as the Ruler Fucto but over all rmes. 1

14 Fgure 3: Tragular Reresetato of the Dvsor Fucto Theorem 8. Let τ be the Dvsor Fucto. The frst four colums of the geeralzed bomal coeffcets of τ, ( for 0 k 3, are etrely tegral. k τ Proof. We kow ( 0 τ ad ( are tegral because of Corollary 9. 1 τ The secod colum s ca be reduced to ( 1 τ( τ 1 τ( 1 τ( τ(τ( 1 τ(1τ( τ(τ( 1. Ths meas that f τ(τ( 1 s odd, the ( s ot tegral. We kow that τ( s τ odd f ad oly f s a erfect square. Therefore, τ(τ( 1 s odd f ad oly f ad 1 are both erfect squares. But t mossble for two erfect squares to be cosecutve so τ(τ( 1 s always eve ad ( s tegral for all. τ 13

15 The thrd colum ca be smlarly reduced to ( 3 τ 1 τ( 3 1 τ( 3 1 τ( τ(τ( 1τ( τ(1τ(τ(3 τ(τ( 1τ(. 4 Oce aga, we kow that τ( s odd f ad oly f s a erfect square. The smallest dfferece betwee two erfect squares s the dfferece betwee the two smallest erfect squares 1 ad 4, whch s a dfferece of 3. Therefore, τ(τ( 1τ( always cotas at least two eve terms ad s always a multle of 4. So ( s always tegral. 3 We ca also look at the Sum of Dvsors Fucto (OEIS A00003 [6] whch s defed as σ( d d ad has the followg roertes: τ 1. Multlcatve wth σ( e e σ( s odd f ad oly f s a square or twce a square. If ɛ 1 for 0 k ad ɛ 0 for k, ( ( ( σ( +1 ɛ m σ σ( 0 ( ( + ɛ ( k+ 1 k+1 1 ( k Ths fal value also must be tegral because we kow ( m σ 8 s codto s satsfed. s tegral wheever Corollary Ths fucto, lke the fuctos that wll follow, turs out to be dffcult to work wth. For examle, determg whe σ(σ( 1 s a multle of 3 s requred for determg whe ( s tegral. The tragular reresetato of σ, show fg. 4, turs out to be m σ really terestg though, amely the arabola of black dots. Determg why ths arabola s formed would be a terestg roject for the future. 14

16 Fgure 4: Tragular Reresetato of the Sum of Dvsors Fucto 3.3 Products of Exoets of the Prme Factorzato of Ths fucto (OEIS A [7], whch we wll deote as ρ, s descrbed ts ame. As a examle, ρ( because the rme factorzato of 3600 s ad The multlcatve defto s ρ( e e. Oce aga, f ɛ 1 for all 0 k ad ɛ 0 for > k, usg ES Theorem 1 we obta ( m ρ ( ( + 1 >0 ɛ ( ρ( 1 ρ( 0 ( k + 1 k (1 1 (k + 1. As ca be see fg. 5, the frst few colums of ( are etrely tegral. Ths s fact m ρ credbly easy to show. Theorem 9. The frst four colums of the geeralzed bomal coeffcets of the roduct of exoets of rme factorzato of fucto, ( for 0 k 3, are etrely tegral. k ef 15

17 Fgure 5: Tragular Reresetato of ρ Proof. Corollary 9 covers ( 0 ρ ad ( 1 ad ρ(1ρ(ρ(3 1 resectvely. etrely tegral. ρ. The deomators of ( ρ ad ( 3 ρ are ρ(1ρ( 1 are ρ Therefore, the thrd ad fourth colums of ( m 3.4 Number of Squares mod The Number of Squares mod fucto (OEIS A0004 [8], or the umber of quadratc resdues mod, couts the umber of uque values the set {x mod 0 x }. Let us deote ths fucto as q. As a examle, q(10 6 because {x mod 0 x } {0, 1, 4, 9, 6, 5, 6, 9, 4, 1, 0} whch has 6 uque values. The multlcatve defto s e +, q( e 6 e+1 + 1, > + whch ca oce aga be used cojucto wth ES Theorem 1 to determe a form for ( m f q ɛ 1 for ɛ 1 for all 0 k ad ɛ 0 for > k: 16

18 ( m q ( ( q( +1 ɛ q( 0 ( +1 + ɛ 6 ( ( ɛ > ( k ( k + 6 > ( k ( k > Sursgly, ths value s always tegral because of Corollary k+ + k Fgure 6: Tragular Reresetato of q(. The comlcated ature of ths fucto makes t extremely dffcult to fgure out geer- 17

19 alzatos of ( m q. For examle, ( q q(q( 1 q(q( 1, so determg whe q( s q(1q( eve wll tell us whe ( s tegral. However, fgurg out whe q( s eve s already q dffcult, ot to meto dealg wth floor fuctos the multlcatve defto. Walter D. Stagl rovdes a dfferet defto of the quadratc resdues hs aer Coutg Squares Z [9]. He defes quadratc resdues to be the elemets Z that are relatvely rme to ; so, q( s the umber of these kds of quadratc resdues. Ths fucto s also multlcatve ad Stagl dscovered the Multlcatve defto of ths q( to be 3,, 3 q( e e e 1, > wth q( q(4 1. Ths fucto turs out to be dvsble because q( r q( r+1 r 3 r ad q( r q( r+1 r r 1 r+1 r r 1 ( 1 r ( 1. Therefore, by ES Theorem, ( s tegral for all ad m for Stagl s q fucto. m q 18

20 Refereces [1] Edgar, Tom ad Mchael Z. Svey. Multlcatve Fuctos, Geeralzed Bomal Coeffcets, ad Geeralzed Catala Numbers. Prert. [] Dudley, Uderwood. Elemetary Number Theory. W. H. Freema ad Comay: Sa Fracsco, CA.. 38, [3] Erds, P. ad R. L. Graham Old ad New Problems ad Results Combatoral Number Theory. Geeva, Swtzerlad: L Esegemet Mathmatque Uverst de Geve, Vol. 8, [4] OEIS Foudato Ic. (011, The O-Le Ecycloeda of Iteger Sequeces, htt://oes.org/a [5] OEIS Foudato Ic. (011, The O-Le Ecycloeda of Iteger Sequeces, htt://oes.org/a [6] OEIS Foudato Ic. (011, The O-Le Ecycloeda of Iteger Sequeces, htt://oes.org/a [7] OEIS Foudato Ic. (011, The O-Le Ecycloeda of Iteger Sequeces, htt://oes.org/a [8] OEIS Foudato Ic. (011, The O-Le Ecycloeda of Iteger Sequeces, htt://oes.org/a0004. [9] Stagl, Walter D. Coutg Squares Z. Mathematcs Magaze. October,