# Bozo~s Lemma

### About: znuz is znees - vol 4 about: zz vol 4 - large cover design

Chapter 7 of volume 4 of Znuz is Znees & on this page of the Tai Shu credit Vidar with a lemma on the sum of two powers:
The nth root of sum of two nth powers is always less than the (N-P)th root of sum of (N-P) powers of same two terms .

Nice credit from GW for his proof of Fermat's Last Theorem. You might describe some day how that all came about. You might also comment as to whether you think he proved the theorem. ### Tags

1. lemma of bozo
2. fermat

### Comments

Seth says
i seem to remember making a obvious observation about roots of powers of the same number and then GW grabbing it and making a big deal about it and naming it after me.  its hard for me to remember the actual details but what you have written above seems similar.  i don't remember much detail of GW's proof but i think it may have been trying to locate the root of Z between an upper and lower bound ... hence the p in your formula above.  i don't remember being convinced myself that he had found the proof of fermat.

strangely enough from that same day i did discover (and actually trivellay proved) what i would have liked to call the first order version of fermat ... that is  x + y = z, where all numbers are integers ... you will never pass a prime factor of x or y into their sum z.  try it, see if you can do it ... the prime factors will alway get filterd out in the sum .   but GW ignored that so called discovery and jumped on the triveal observation ... which apparently he was able to prove.

Seth says
i thought i had blogged that here but i cant find it.   let me restate it:
• assume x, y, and z are integers
• assume that there are no prime factors that are common to all x, y, and z (in other words all common factors have been already factored out)
• then if x + y = z
• any prime factor of either x or y, will not be a prime factor of z i remember being able to prove that, but that was in my youth ... the proff escapes me at the moment.

for example:  2*3 + 5*7 = 41 ... but 41 does not contain 2, 3, 5, or 7 as a factor.

incidentally the reason i called this the first order fermat is that it seems to me that there is a commonality between roots and factors ... fermat is about integer roots being filtered out of a sum ... mine is about integer factors being filtered out of a sum.

### See Also

1. Thought Fermats Last Theorem with 1 viewings related by tag "fermat".
2. Thought about: from a flat triangle to fermats last theorem with 0 viewings related by tag "fermat".