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 6 + 4 b + 6 b + b 7 + 6 b +0 b + 4b 8 + 7 6 b + 4 b +0 b + b 4 9 + 8 7 b + b +0 b + b 4 0 + 9 8 b +8 6 b + 4 b + b + b + 0 9 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) 4 0 0 form the Lening Tower of Pscl, / / / / 6AD / / / / / 6 0 Omr / / / / Khyyám, / / / / / / 00AD / / / Al / / Krji, / / / / 000AD / / / Pingl, 00BC 4
There re infinitely mny prticulr cses: + b 4 + b + b + 4 b + b 6 + 4 b + 6 b + b For exmple, =, b = gives the nturl numbers (A00007) 0,,,, 4,, 6, 7,... + b 4 + b + b + 4 b + b 6 + 4 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) + 4 + b + b + 4 b + b 6 + 4 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
=, 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 7 7 4 9 (convergents to the continued frction) to the squre root of 99 70 9 69 77 408 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 : 77 408.,, 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.
=, 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
= 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 9 ++4608x 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 = 6 + 4 b+6 b +b P 8 = ( +b)( 4 +4 b+4b ) U n (cosθ) = sin(n+)θ sinθ P 9 = ( +b)( 6 +6 4 b+9 b +b ) P 0 = ( 4 + b+b )( 4 + b+b ) P = 0 +9 8 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
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