Problem:
I was wondering if anyone could help me with finding a continous
function (with 5 or fewer constants) that will generate the series
35, 45, 60, y , 120, 180, 280, 450, 744, 1260 as x assumes the values
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
SPOILER
SPOILER
From - Thu Feb 12 10:17:35 1998
From: pmontgom@cwi.nl (Peter L. Montgomery)
Subject: Re: Number Problem for Experts
Organization: CWI, Amsterdam
> I was wondering if anyone could help me with finding a continous
> function (with 5 or fewer constants) that will generate the series
> 35, 45, 60, y , 120, 180, 280, 450, 744, 1260 as x assumes the values
> 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
> Also, i need to find the value of y in terms of standard functions.
All prime divisors of the supplied numbers are below 10,
except the divisor 31 of 744. I observe that 31 = 2^5 - 1,
and that 2^6 - 1 = 63 divides 1260. This pattern persists
in the other direction, since 15 divies 450 and 7 divides 280.
Look at the quotients y/(2^(x-4) - 1):
x 1 2 3 4 5 6 7 8 9 10
y 35 45 60 ? 120 180 280 450 744 1260
2^(x-4)-1 -7/8 -3/4 -1/2 0 1 3 7 15 31 63
quotient -40 -60 -120 ? 120 60 40 30 24 20
The largest prime divisor in the quotient row is 5.
There is a simple formula for this row, which I'll leave to the reader.
If we extend our formula to non-integral values,
we guess an irrational value near 83 for y when x = 4.
--
Peter-Lawrence.Montgomery@cwi.nl San Rafael, California
A mathematician whose age has doubled since he last drove an automobile.