PROPERTIES OF THE POSITIVE INTEGERS

Transcription

1 PROPERTIES OF THE POSITIVE ITEGERS The first itroductio to mathematics occurs at the pre-school level ad cosists of essetially coutig out the first te itegers with oe s figers. This allows the idividuals to slowly recogize the abstract character of whole umbers give by the sequece- R={1,2,3,4,5,6,7,8,9,10,11,.2,2+1} with =1,2,3,etc The kider-gardeers soo recogize that two apples plus three apples makes five apples ad that apples ca be replaced by aythig else ad the aswer will still be five of the aythig else. As the years pass the elemetary school childre lear about the cocepts of additio, subtractio, divisio, ad multiplicatio of itegers ad o-itegers. Certaily by the time they reach middle school they will be able to maipulate collectio of itegers to obtai equivalet expressios. They will also recogize the differece betwee eve (2) ad odd (2+1) itegers ad the cocept of zero. By the time high school rolls aroud studets will have mastered the cocepts of algebra ad itroductory calculus all arisig from their early experiece of umber coutig by figers. I additio may of the high school studets will have oted that the itegers ca be broke ito prime umbers ad composite umber. A prime umber refers to ay iteger which ca be divided oly by itself ad oe while a composite umber cosists of the product of several primes take to specified powers. The above groupig of itegers breaks ito the prime sequece- P={2,3,5,7,11,13,19,23,29,31,37,41,43,47, } ad the composite sequece- C={0,4,6,8,9,12,14,15,16,18,20,21,22,24,25,26, } If we eglect the first two itegers i the P sequece, we ote that all the remaiig itegers are odd ad have the geeric form 6+1 or 6-1, so that for istace 41=6(7)-1 ad 19=6(3)+1. Thus oe ca state that A ecessary but ot sufficiet coditio that a umber >3 be a prime is that 61 The reaso this law is ot sufficiet is that certai composite umbers such as 25 also satisfies 6(4)+1. A improved way to idetify primes is by meas of the umber fractio- { sum of f ( ) all divisors of excludig ad 1]} [ ( ) 1] We first foud this poit fuctio about a decade ago. It is easy to evaluate sice the sigma fuctio () of umber theory is a well kow quatity i most mathematics computer

2 programs. The iterestig property of f() is that it vaishes oly whe is a prime, but remais positive whe is a compoud umber. Because of the presece of i the deomiator of its defiitio its value remais o average below f()=1. The few f()s which exceed f() of about 1.4 I have termed super-composites. They typically cotai products of 2 ad 3 take to specified powers. A graph of the umber-fractio i the rage 10<<80 follows- We see that oly the prime umbers have vaishig f(). The remaiig poits are either super-composites such as =48, 60, ad 72, or they are composites composed of two(semiprimes), three, etc prime products makig up. The value of f(35)=12/35 idicates that 35=5 x7 is a composite(semi-prime) cosistig of the product of two primes 5 ad 7 take to the first power each. The umber =2 is always a composite sice f(2 )=[1-1/2-1 ]. It will ever become a super-composite sice it approaches oe as goes to ifiity. A slight variatio o this umber produces the Mersee umbers- 2 1 which yields [ (2 1) 2 ] f ( ) (2 1) O settig =17 we get = ad f()=0. So is a prime. Goig to the much larger umber =

3 we agai fid f()=0 so it is also a prime umber. Pierre Fermat proposed i the sixtee hudreds that the umber F=2( 2^ )+1 is a prime umber for all positive. Leoard Euler however proved him wrog for the value of =5, although the =1 through 4 clearly are primes. Euler struggled with this problem for moths fially beig able to actually factor F= ito its compoets. Usig the umber fractio we ca fid the composite ature of F at =5 simply by otig that [ (2 1) 2 2] f ( 2 1) (2 1) Sice f() does ot vaish it must be a composite. ote to prove this we actually ever eeded to fid the semi-primes compoets =(641)( ) As already metioed, super-composites have the form- =2 a 3 b 5 c where a>b>c ad a f()>1.4. Such umbers become particularly cospicuous whe gets large. We show you here a example for the super-composite =17280-

4 ote here that this super-composite stads head ad shoulders above its immediate eighbors. There are o primes i the rage show, however, there are multiple semiprimes such as 17273=23x751,17279=37x467,17281=11x1571 ad 17287=59x293. Also we have composites such as 17286=2x3x43x67 cosistig of more tha two products of primes. We ext tur to the summatio of the first itegers. Startig with =1 we get the partial sums- Partial sum: First differece: Secod differece: This implies that the sum will be a quadratic i of the form S()=A+B+C 2. Matchig the first three partial sums the yields A=0, B=1/2, ad C=1/2. Hece we have- S()=0+/2+ 2 /2=(+1)/2 For just the odd umbers we get the partial sums- Partial sum: But these are recogized at oce to be the square of the itegers. Hece we fid- S(99)=[(99-1)/2] 2 =49 2 =2401 So the sum of all odd itegers through 99 is 2401 ad thus approximately half of the sum of all itegers through 99 which have the higher value of It leaves us with the sum of the eve itegers through 99 to be =2549. A iterestig observatio is that the sum of the reciprocals of the itegers reads R( ) For fiite this sum has fiite positive values ad oe would at first glace predict that a fiite positive value should remai as goes to ifiity. This turs out however to ot be the case for oe has- R ( ) 1 1

5 This seies is kow as the harmoic series. That it diverges follows from the fact that 1 0 x1 dx x l( ) So sice l() is ifiite as goes to ifiity so must R() be. There is aother type of ifiite series kow as the geometric series. It reads- S(r)=1+r+r 2 +r 3 +.= 0 r where r is a fractio lyig i o<r<1. This series ca be summed i closed form by subtractig rs(r) from S(r) to yield- S(r)=1/(1-r) If the upper limit of the sum is kept at the fiite value of =, we fid that- r (1 r (1 0 r 1 ) ) So for the special case of r=1/2 ad =5 we get (2 6-1)/2 5 =63/32. ext let us look at the product of the itegers- 1x1=1 1x2=2 1x2x3=6 1x2x3x4=24 1x2x3x4x5-120 The terms o the right of this equality are kow as the factorials so that 6=3!, 24-4!, ad 120=5!. Furthermore we have the idetity that ( +1)! =(+1)! so 6!=6x120=720. Aother iterestig idetity ivolvig factorials is that-

6 ! t 0 t exp( t) dt This fact allows us to evaluate itegrals of the form- x x0 a a! exp( bx) dx b a1 There are certai algebraic expressios such as- y 2 =1+k 2 x 2 which have solutios for iteger values of x ad y provided k 2 is a iteger. These are kow as Diophatie Equatios. Take the special case of k=sqrt(2);this produces the positive iteger pair- [x,y]=[2,3],[12,17],[70,99],[408,577] ad so o. Solvig the Diophatie equatio we get- 2 y ( ) 1 O(1/ y ) x y 2 y 8y 16y whe expaded as a series. We ca take ay of the Diophatie solutios give above to have a covergig series for sqrt(2). Takig [x,y]=[408,577] we get the very rapidly coverget series O (1/ (322929) 8(322929) Already the first three terms of the series multiplied by 577/408 yield the approximatio- sqrt(2) compared to the exact value of- sqrt(2)= This approximatio is good to 16 places after the decimal. Aother property of the positive itegers is that there are certai which match the sum of all its divisors mius. These are called the perfect umbers. I terms of () ad f() we ca express them as- 4

7 2=() or f()=1-(1/) The aciet Greek mathematicia Euclid showed that a ecessary coditio for a umber to be perfect is that it satisfy- 2 1 (2 1) for certai primes It ideed geerates such umbers for =2,3,5,7,13,17,19,31,61,89, but fails for for =11,23,29,37,41, 43,47, etc.. Lookig at the values of where ()-2=0, we fid the first eight perfect umbers to be- 6, 28, 496, 8128, , , , ote that each of these perfect umbers are eve. Fially we look at the cocept of logarithms for itegers. I a decimal iteger system we ca defie ay iteger as- =10 a with a the expoet of 10 also referred as the logarithm of to base 10 Each iteger will have associated with it a uique logarithm which most of the time will be a o-iteger. Here is a brief table of versus a=log() a=log() The advatage of logarithms i the past was that they could speed up the multiplicatio ad divisio of large umbers. Today they are o loger ecessary sice direct calculatios with electroic calculators ca do the job faster. This is also the reaso slide rules have bee replaced by had calculators. A typical multiplicatio usig logarithms is- 3x x More extemsive tables will brig the result 12. There are two bases commoly used for logarithms. The first of these is the Briggs logarithm which uses the base 10 ad is desigated by log() i the literature. The other is the atural logarithm based o the base e= ad desigated as l(). The two logarithms relate to each other by the multiplicatio factor l(10)= thus- l(5)=l(10) log(5)= x0.6989= For ay other base b we have-

8 =exp(l()=b log b () So for b=2 we get log 2 (8)=3 or the equivalet 2 3 =8. U.H.Kurzweg July 27, 2018 Gaiesville, Florida