1. ## Euler's Conjecture

Euler's Conjecture

Euler made a conjecture that could not be valid for x,y,z.w,n whole numbers (positive integers) as a generalization of Fermat's Theorem.

I would re-write that as as a generalization of my proof of Fermat's Theorem. (Vectors were later. So was Special Theory of Relativity. In case you hadn't heard), so x,y,z and w are independent variables in the vector space (x,y,z). One would then relate them by scaling factors and try to find a relation true for all variables, (xt,yt',zt"), wt'", as in my proof of Fermat's theorem via Special Relativity applied to numbers independently of physical interpretation.

I am not familiar with the conjecture (or why Euler thought it was reasonable) or its negation by computer... That said, however, I would be very suspicious of the latter for all sorts of technical reasons. And also (especially) because it ignores the distinction between radial and Cartesian coordinate systems.

This would be the equation of a hypersphere. If proven, it would then be true for all whole numbers automatically. Maybe someday I'll work on it. After I relate the Pauli/Dirac matrices to the Higgs equation and string theory....
-------------------------------

Of course, if Fermat's theorem is true, then by my proof is not an integer (it is at least irrational - i.e., not divisible by an integer).

Consider the equation Since is not an integer (i.e. not divisible by an integer), then if is an integer, cannot be an integer.

QED

You heard it here first...

So this "proof" is related to Fermat's Theorem. I don't trust disproof by computer... there is too much that can go wrong in the very high digits...... But it is not ironclad for me, since I am not focusing on it - others are welcome to try if they can understand it and not just quote articles from Popular Mathematics about it....

2. ## Re: Euler's Conjecture

None of that logically follow and stop addingall those superflous i,j and k, they do nothing but add confusion.

3. ## Re: Euler's Conjecture

Originally Posted by BuleriaChk
Euler's Conjecture

Euler made a conjecture that could not be valid for x,y,z.w,n whole numbers (positive integers) as a generalization of Fermat's Theorem.

I would re-write that as as a generalization of my proof of Fermat's Theorem. (Vectors were later. So was Special Theory of Relativity. In case you hadn't heard), so x,y,z and w are independent variables in the vector space (x,y,z). One would then relate them by scaling factors and try to find a relation true for all variables, (xt,yt',zt"), wt'", as in my proof of Fermat's theorem via Special Relativity applied to numbers independently of physical interpretation.

I am not familiar with the conjecture (or why Euler thought it was reasonable) or its negation by computer... That said, however, I would be very suspicious of the latter for all sorts of technical reasons.
2682440^4 + 15365639^4 + 18796760^4 = 20615673^4

You could do that by hand, in a few minutes.

Both sides are equal to 180630077292169281088848499041
And also (especially) because it ignores the distinction between radial and Cartesian coordinate systems.

This would be the equation of a hypersphere. If proven, it would then be true for all whole numbers automatically. Maybe someday I'll work on it. After I relate the Pauli/Dirac matrices to the Higgs equation and string theory....
-------------------------------

Of course, if Fermat's theorem is true, then by my proof is not an integer (it is at least irrational - i.e., not divisible by an integer).

Consider the equation Since is not an integer (i.e. not divisible by an integer), then if is an integer, cannot be an integer.

QED

You heard it here first...

So this "proof" is related to Fermat's Theorem. I don't trust disproof by computer... there is too much that can go wrong in the very high digits...... But it is not ironclad for me, since I am not focusing on it - others are welcome to try if they can understand it and not just quote articles from Popular Mathematics about it....
ETA: The fourth root of (2.7^4 + 15.4^4 + 18.8^4) is 20.63, not bad

4. ## Re: Euler's Conjecture

Apparently, the smallest counter-example was found in 1988, the only example where the values are less than a million:

95800^4 + 217519^4 + 414560^4 = 422481^4

In the late 70's, our university had just installed the latest computers from CDC, and the school departments were given provisional accounts to test the capabilities. I started a search for this example, just to see how the usage accounting worked. I was surprised when it showed hardly any usage, so I continued to run the program. Turned out they hadn't turned on the report to user feature, and after the test period, my program accounted for over 40% of the total university usage. And the department allocations were based on that! The math department was very grateful, my friends in the cosci dept were amused.

ETA: It all started one night when I was going to sleep and I wondered about sums of cubes, so I started with 3 and 4 and 5, naturally. I was amazed when the sum of their cubes turned out to be 216, or 6 cubed.

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