The Leaning Tower of Pingala

Transcription

1 The Lening Tower of Pingl Richrd K. Guy Deprtment of Mthemtics & Sttistics, The University of Clgry. July, 06 As Leibniz hs told us, from 0 nd we cn get everything: Multiply the previous line by nd dd b times the line before tht + b 4 + b + b + 4 b + b b + 6 b + b b +0 b + 4b b + 4 b +0 b + b b + b +0 b + b b +8 6 b + 4 b + b + b b +6 7 b +6 b + b 4 + 6b 0 If you tip your hed on one side 6 you ll see tht the coefficients (A097) form the Lening Tower of Pscl, / / / / 6AD / / / / / 6 0 Omr / / / / Khyyám, / / / / / / 00AD / / / Al / / Krji, / / / / 000AD / / / Pingl, 00BC 4

2 There re infinitely mny prticulr cses: + b 4 + b + b + 4 b + b b + 6 b + b For exmple, =, b = gives the nturl numbers (A00007) 0,,,, 4,, 6, 7,... + b 4 + b + b + 4 b + b b + 6 b + b =, b = gives the Mersenne numbers (A000) 0,,, 7,,, 6, 7,... n Everyone believes tht infinitely mny of them (n =,,,7,...) re prime...but no-one cn prove tht! =, b = gives the Fiboncci numbers (A00004) b + b + 4 b + b b + 6 b + b 0,,,,,, 8,,, 4,, 89,... bout which whole books hve been written. They re the numbers of wys of pcking dominoes in (n ) box: b

3 =, b = gives the Brhmgupt-Pell numbers (A0009) 0,,,,, 9, 70, 69, 408,... These were probbly known to the Bbylonins nerly 4000 yers go. They re the denomintors of good pproximtions (convergents to the continued frction) to the squre root of Suppose tht you wnt to know if there s number whose squre is. is too smll, nd is too big. So tke the verge,. Divide it into, giving 4. is too big, 4 is too smll. We ve lredy lerned tht the rithmetic men is greter thn the geometric men! Tke the verge of nd 4 : 7. Then the verge of 7 nd 4 7 : ,, 7, 77, re the st, nd, 4th, 8th of the convergents; 408 nd = +, 7 = +, 77 = 408 +,... The process doesn t stop!! is irrtionl!! In Bbylonin = ; 4,, 0, 7, 46, 6, 4,... compred with 77/408 = ; 4,, 0,, 7, 8, 4,... They knew tht ; 4,, 0 is better thn ; 4,, nd, tht if they hd enough cly tblets, they could get s close s they liked.

4 =, b = gives the Jcobsthl numbers (A0004) 0,,,,,,, 4, 8, 7, 4,... which, prt from the zeroth, re ll odd. In fct J n+ = J n +( ) n They were useful to us when we nlyzed Conwy s subprime Fiboncci sequences [Mth. Mg., Dec. 04.] They re lso the number of wys of tiling n rectngle with nd squre tiles. Or the number of wys of tiling n rectngle with dominoes nd squres. These fcts follow from the following digrms: J n } J n J n = J n +J n J n J n } J n J n = J n +J n J n 4

5 = x, b = gives the Chebyshev polynomils of the first kind, T n (x), =x, b = gives Chebyshev polynomils of the second kind, U n (x); (A0490 nd A0999), U 0 (x) = U (x) = x U (x) = 4x U (x) = 8x 4x U 4 (x) = 6x 4 x + U (x) = x x +6x U 6 (x) = 64x 6 80x 4 +4x U 7 (x) = 8x 7 9x +80x 8x U 8 (x) = 6x 8 448x 6 +40x 4 40x + U 9 (x) = x 9 04x 7 +67x 60x +0x U 0 (x) = 04x 0 04x 8 +79x 6 60x 4 +60x U (x) = 048x 0x x 7 79x +80x x which stisfy the following formuls: ( x )U n xyu n +n(n+) = 0, Let s fctor our originl polynomils: P = P = +b P 4 = ( +b ) P = 4 + b+b P 6 = ( +b)( +b ) P 7 = b+6 b +b P 8 = ( +b)( 4 +4 b+4b ) U n (cosθ) = sin(n+)θ sinθ P 9 = ( +b)( b+9 b +b ) P 0 = ( 4 + b+b )( 4 + b+b ) P = b+8 6 b + 4 b + b 4 +b P = ( +b)( +b)( +b)( 4 +4 b+b ) The underwved polynomils re primitive prts, nlogous to the cyclotomic polynomils. This illustrtes tht our sequences re divisibility sequences, tht is: m n implies tht u m u n Here s how to see tht: u n = u n +bu n. Guess tht u n = Ax n. Ax n = Ax n +bax n x = x+b x = ± D where D = 4b sy x = α or β, so tht u n = Aα n +Bβ n nd u 0 = 0 nd u = give 0 = A+B, = Aα+Bβ, A = B = /(α β) nd the divisibility is cler. u n = αn β n α β

6 The Lucs-Lehmer theory tells us tht prime p divides u p ( D p) ( ) D where p is the Legendre symbol: ± ccording s D is, or is not, qudrtic residue (squre) mod p (or is zero if p D). For exmple, for the Fiboncci numbers the discriminnt D =. So u p is divisible by p if p is of shpe 0k ±, nd u p+ is divisible by p if p is of shpe 0k ±, nd u n is divisible by. But this is not only if!! For exmple u 4 (= 77) nd hence u 4k for ll k, but in fct U 7k for ll k. ( ) D We know tht the rnk of pprition of p is divisor of p p, but we don t know which! Here s something else we don t know! A member, u n, of one of these sequences cn be prime only if n is prime; since u pq is divisible by u p nd by u q. But if p is prime, then u p is not necessrily prime! Among the Fiboncci numbers u =, u =, u 7 =, u = 89, u =, u 7 = 97 re ll prime, but u 9 = 48 = 7 is not! We do not even know if there re infinitely mny Fiboncci primes,... or infinitely mny Mersenne primes,... or infinitely mny Brhmgupt-Pell primes,... or infinitely mny Jcobsthl primes, There re infinitely mny things we don t know!! but there re infinitely mny things WE DO KNOW!! Tht s the beuty of Mthemtics!! 6