SOME NEW OBSERVATIONS ON MERSENNE NUMBERS AND PRIMES

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Angelina Hines
5 years ago
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1 SOME NEW OBSERVATIONS ON MERSENNE NUMBERS AND PRIMES Euclid observed several thousad years ago i his boo The Elemets that - 1+=3 1++4= = = = = =55 I this set of equatios oe sees that may of the sums icludig 3, 7, 31, 17 are rime umbers. I moder otatio for a fiite geometric series of this tye oe has- S [ N ] N 0 N 1 1 Here S[N] is a rime umber for N=1,,4,6,.. but fails to be rime for N=3,5.7,.. The Frech riest ad mathematicia Mari Mersee( ) observed from Euclid s results that -1=3, 3-1=7, 5-1=31, ad 7-1=17 are all rime umbers. This led him to the coclusio that- Ay umber of the form M[ ]= ^( )-1, where is the th rime umber, is liely to be a rime Today oe refers to the M[ ]s as Mersee Numbers ad so far 48 of these have bee foud to be rime. The majority of the M[ ] s, however, are comosite umbers such as 11-1=047=3 89 ad 3-1= = The larger the rime the larger will be the value of M( ). It is our urose here to re-examie the Mersee Numbers ad Primes to see if oe ca come u with some ew roerties heretofore urecogized. Our startig oit is to ru the followig MAPLE search rogram- for from 1 to 18 do {, ithrime(), ^ithrime()-1, isrime(^ithrime()-1)}od; This yields the results-

2 Iteger, Prime umber, Value of M[ ]=^-1 Prime (P)or Comosite(C) 3 7 P P P C P P P C C P C C C C C C P We see from these results that the size of Mersee Numbers ad Primes icrease raidly with icreasig, ad that they always ed i 1 or 7. Also oe otices that M[ ] mod(6) is always equal to 1. Thus ( )mod(6)=1, 1571 is a rime, but is a comosite.the biary versio of all M[ ]s cotai oly 1s. Thus =17 i decimal reads i biary. We have lotted the logarithm of these Mersee Numbers ad Primes i the grah below. Note that a excellet aroximatio to the value of a Mersee Number for larger is give by M[]=ex{l()* } sice for larger s the term ^( ) is much larger tha 1.

3 Let us loo at the first seve Mersee Primes i this table ad write dow the differeces from their earest eighbor. The results are summarized i the followig table- M[ ] M[ +1 ]-M[ ] = =336 4=84 96= =510 4=180 96=30 384= = = = = = = = = = = = = = = = = = = = = = = = = = = = =

4 You will otice that the differece betwee eighborig Mersee Numbers, be they rime or comosite, will always be a iteger multile of 4. This umber was recetly foud by us whe we observed that all Mersee Numbers M[] lie alog a diagoal i the 4 th quadrat of a iteger siral defied i olar coordiates as [r,θ]=[,π/8] ad alog a diagoal i the 1 st quadrat of a itegral siral defied as [r,θ]=[,π/6]. The sacig betwee odd umbers alog the secified diagoals are 8 ad 6, resectively. Hece the lowest umber divisible by both 6 ad 8 is 4 ad this leads to the coclusio that- {M[ m ]-M[ ]}/4=iteger for 3 <m I terms of modular arithmetic, this statemet is equivalet to sayig- {M[ m ]-M[ ]} mod(4)=0 We ca calculate the value of the above iteger by otig that- m ( m ) {{( 1) ( 1)}/ 4 ( 1) / 4 iteger To test these results cosider the Mersee Numbers M[11]= 11-1 ad M[313]= Oe fids- with- [( 313 1) ( 11 1)]mod(4) 11 ( 10 1) mod(4) 0 iteger= 11 ( 10-1)/4= Oe otices from the last table that the sacig betwee M[ m ] ad M[ ] as icreases will icrease to a ever larger sacig of 4(4 ), where the iteger is 1 for 5, for 7, 5 for 13 ad 7 for 17. A estimate for the sacig is about 4(4^( /))=4(^( )). The actual sacig G[ ] betwee the rimes M[107] ad M[17] is foud to be- G[107]= ((^17-1)-(^107-1))= We ca write the value of G[107] as - G[107]= (4 5 )= The factor 4(4 5 ) is see to lie i the ball-ar of the sacig estimate of 4( 107 ). Sice all Mersee Numbers ad Primes are of the form 6+1, we ca distiguish comosite(c) from rime (P) forms of M[ ] by looig at-

5 ( M[ ] m 6 ) for 3,5,7,11,... sqrt( M[ ]) If a iteger solutio for m exists the M[ ] is comosite. Otherwise it will be rime.(see our earlier df file o the Sieve of Eratosthaes of why this is so). A more elegat way to exress this last equatio is to say that M[ ] is rime if- { M[ ] }mod(6 ) 0 for 3,5,7,11,.. sqrt( M[ ] If we loo at M[11]=047, we fid that =3 yields zero i the mod oeratio. Hece the Mersee Number M[11]=3{3+6(11)}=3 89 is a comosite. O the other had, M[13]=8191 roduces o zero mod i 3< <89 ad so is rime. This test is just as useful as the classic Lucas-Lehmer test ad has the advatage that the factors for a comosite umber aear directly ad that it wors for all odd umbers of the form N=6+1. The largest Mersee Prime ow as of February 013 is- M[ ]= This umber is oly the 48th Mersee Prime foud so far. It is susected that there are a ifiite umber of such rimes but tryig to fid more of these aears to be a rather urewardig effort similar to calculatig π beyod a billios laces. The M[ ] rimes are useless for alicatios i crytograhy sice they are so well ow, ad because they mostly exceed i digit legth aythig which could be of use for ublic eys. U.H.Kurzweg July 013

Perfect Numbers 6 = Another example of a perfect number is 28; and we have 28 =

Perfect Numbers 6 = Another example of a perfect number is 28; and we have 28 = What is a erfect umber? Perfect Numbers A erfect umber is a umber which equals the sum of its ositive roer divisors. A examle of a erfect umber is 6. The ositive divisors of 6 are 1,, 3, ad 6. The roer