CONSTRUCTING TRUNCATED IRRATIONAL NUMBERS AND DETERMINING THEIR NEIGHBORING PRIMES It is well kow that there exist a ifiite set of irratioal umbers icludig, sqrt(), ad e. Such quatities are of ifiite legth ad have the roerty that they ca ever be rereseted as the ratio of two real umbers. Segmets of irratioal umbers of fifty digit legth or so are of iterest i coectio with ublic key crytograhy (RSA) where the roduct of two segmets of differet irratioals adjusted to be rime ca lead to early ufactorable semi-rimes N=q. It is our urose here to exted several of our earlier studies o semi-rimes to geerate some large segmets of irratioal umbers lus some related semi-rimes usig fiite segmets of the roducts of some of the better kow irratioals. Our startig oit is to reset a few the better kow irratioal umbers i terms of their basic defiitios. We have ex() e.7888845945...! ( ) 4 3.459653589793... ( ) l()... J () - t ( ).6934785599453... ( ) 5.68339887498948... (!).4435637395....79585333667... l( t)ex( t) dt.577566495386... These costitute the best kow ad ofte used irratioal umbers occurrig i the mathematical literature. They are art of a ifiite set icludig such additioal irratioals as-
( ) ( )... Here is a root of ad lies ear. Thus the square root of five, which aears i the defiitio of the golde ratio above, is give by the cotiued fractio- 5.36679774997896964... 4 4 4... It is imortat to recogize that segmets of such irratioals eed ot ecessarily have their digits eter i a radom maer. If they did the all digits betwee ad 9 would aear equally. Let us demostrate this o-radomess by lookig at the first 5 terms of ex() ad. Here we have- e.788884594535368747356649775747937 =3.45965358979338466433837958849769399375 so that the digit cout looks as follows- digit 3 4 5 6 7 8 9 e 6 3 8 4 5 5 3 8 5 3 5 5 8 5 5 4 4 5 8 The cout shows that the fifty digit log segmets of these two irratioals are ot comletely radom because the aearaces of the te ossible digits are ot five each. The radomess will icrease as the segmet legths ad advace toward ifiity. The fact that fiite legth segmets of irratioals are ot comletely radom should ot iterfere with quickly comig u with some trucated roducts of irratioal umbers ad the rime umb ers located i their eighborhoods. This will make ossible the use of some semirimes for ublic keys roduced with a miimum of effort. I additio ay of these rimes will be idetifiable by short ad uique codes. We choose to defie a fiite segmet of a irratioal umber by the code- K L a ( ) m k k k
Here a k reresets oe of K commo irratioals take to the k th ower. The segmet starts with the + term ad is m digits log. Thissegmet will be desigated by M. Let us look at a secific examle. Take- M 6{ e(4) (/ 3) }5 This code roduces the fifty digit log segmet- M=567978339363999754678559946956 Usig a bit of modular arithmetic, we have M mod(6)=4 which imlies that that M lies alog the radial lie 6+4 i the followig hexagoal iteger siral- We have show i several earlier otes (look at both our MATFUNC ad TECH-BLOG ages for the years -5) that all rimes greater tha 3 must lie alog the two radial lies 6. I the grah these rimes are desigated by blue circles. To fid all rimes i the immediate viciity of M will require we look at- isrime( M+3+6) ad isrime( M++6 ) This is a easy search usig the oe lie rogram- for from - to do {, isrime(m++6)}od;
I a slit secod oe fids that the two closest rimes are M+33 ad M-53. These read- P = 567978339363999754678559946989 P = 56797833936399975467855994693 Here P mod(6)= ad P mod(6)=5. Note that 5 is equivalet to - i the hexagoal diagram. So P will be a blue circle alog the radial lie 6+ ad P oe of the blue cirles alog 6-. As ca be see, it took very little effort to fid the above rimes sice determiig whether or ot a umber is rime is rovided by a very simle comuter evaluatio. I additio we ow have a way to store ad trasmit large rime umbers by a code which itself ca be ecryted. For examle we ca comletely describe the fifty digit log rime P by- P =M+33=6{e(4),(/),(-3)}5+33 Although the roduct of P ad P will ot make a good ublic key N because of their roximity to each other, oe ca easily costruct a coule of rimes searated from each other by orders of magitude. Such a semi-rime N will be almost ubreakable wheever the umber of digits i the Ps is large eough. Let us demostrate this for- M 3 =7{7(/3)(3)e(-)}5 where M 3 mod(6)=3 M 4 ={()J()l(-/)}45 where M 4 mod(6)= I this case we fid the earest rimes to be P 3 =M 3-4= 63559688988896367596354384574844557339 P 4 =M 4 +7= 877476347743974887933536866768567 Oe ca easily verify by comuter that these last two umbers of 5 ad 45 digit legth, resectively, are rime umbers. Their roduct yields the 95 digit log semi-rime- N=557648884349438984846573385487383563963664838774898 76986939646976775697
Without a kowledge of either P 3 or P 4 it is highly ulikely that ayoe icludig our big data Natioal Security Agecy ( NSA) would be successful i factorig this N i ay reasoable amout of time usig eve their latest high seed comuters. Although the RSA aroach to secret message trasmissio is at reset still secure usig fifty or so log rime umbers, this will ot cotiue to hold with time. Loger ad loger rime umbers will be required callig ito questio the efficacy of ublic keys N i RSA crytograhy. It suggests that erhas oe should cosider a ew ad simler aroach to ecryted electroic message trasmissio based o the reset method of usig ecoded forms for certai large irratioal umbers segmets adjusted to be rimes. Oe could evisio a way the roduct of M ad a message S would be set as D=S o the ublic airwaves. If the seder simultaeously seds a secod sigal cotaiig a ecryted form of M which oly he ad a friedly receiver would uderstad, the message will have bee secretly trasmitted. Cosider the case were the message is S=345 ad M is defied as- M=5{7(/)(-/5)}= 55944549933. The ecryted message becomes- D=(M)(S)= 683549366976699479985 There is o way a adversary could deciher D. However the friedly receiver (kowig what M is) will be able to raidly decode thigs as D/M=S=345. UHK Jue, 6