PR 4 # 12

Wacky · 2006-06-12, 16:03

Find unequal rational numbers, a, b, ( other than 2 and 4 ) such that $a^b = b^a$

There are an infinite number of such pairs. Give the general solution.

mfgoode · 2006-06-14, 08:44

Quote:

Originally Posted by Wacky

Find unequal rational numbers, a, b, ( other than 2 and 4 ) such that $a^b = b^a$

There are an infinite number of such pairs. Give the general solution.

Quote:

Originally Posted by Mally

Hint: Euler proved that 2 and the 4 are the only integers satisfying the eqn. So now give the general solution for rationals

Mally

retina · 2006-06-14, 10:04

There is one more set of integers satisfying the equation: {-2, -4}

mfgoode · 2006-06-15, 04:03

Quote:

Originally Posted by retina

There is one more set of integers satisfying the equation: {-2, -4}

I got this info from the book 'Numbers' by David Wells who seldom makes a mistake in the data he presents.

It clearly says ' Euler showed that the only integer solution to a^b = b^a is 4^2 =2^4 =16.'
Unless Wells takes the answer 1/16 with the negative no.s to be a non integer solution. Possibly thats the correct interpretation.

However when it comes to rational numbers the -ve sign cancels out I would say, as is the case with (-2,-4) but I' not sure.

Mally

alpertron · 2006-06-15, 13:52

a = (1 + 1/n)[sup]n[/sup]
b = (1 + 1/n)[sup]n+1[/sup]
n can be any integer different from zero.

retina · 2006-06-16, 02:53

Quote:

n can be any integer different from zero

Actually, n can be any integer except zero or minus one.

2006-06-12, 16:03	#1
Wacky Jun 2003 The Texas Hill Country 3²·11² Posts	PR 4 # 12 Find unequal rational numbers, a, b, ( other than 2 and 4 ) such that $a^b = b^a$ There are an infinite number of such pairs. Give the general solution.

2006-06-15, 13:52	#5
alpertron Aug 2002 Buenos Aires, Argentina 2·683 Posts	a = (1 + 1/n)[sup]n[/sup] b = (1 + 1/n)[sup]n+1[/sup] n can be any integer different from zero. Last fiddled with by alpertron on 2006-06-15 at 13:53

2006-06-14, 10:04	#3
retina Undefined "The unspeakable one" Jun 2006 My evil lair 2×11×283 Posts	There is one more set of integers satisfying the equation: {-2, -4}