Determine all possible integers n less than 10
100 that simultaneously satisfy these conditions:
- n divides 2n
- n - 1 divides 2n - 1
- n - 2 divides 2n - 2
Note: As an extra challenge, determine a semi-analytic (simple calculator + p&p) solution to this problem.
**** Adapted from a problem appearing at a William Lowell Putnam Mathematics Competition
Testing only n up through 2000,
clearvars,clc
for n=sym(1:2000)
pwr2=sym(2)^n;
if mod(pwr2,n)==0
if mod(pwr2-1,(n-1))==0
if mod(pwr2-2,(n-2))==0
disp([n,pwr2])
end
end
end
end
finds only n=4 and n=16 work:
[4, 16]
[16, 65536]
showing n and 2^n
as
4 divides 16; 3 divides 15; 2 divides 14
16 divides 65536; 15 divides 65535; 14 divides 65534
From there on, nothing is found, having checked through n = 2000, though that's a far cry from a googol.
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Posted by Charlie
on 2023-06-15 09:41:33 |
computer exploration
