MHBLast Three Digits of Complex Number Sum and Product Equations
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The discussion revolves around finding the last three digits of the expression \( u^4 + v^4 + x^4 + y^4 + 4uvxy \) given specific conditions for the complex numbers \( u, v, x, y \). Participants utilize Newton's identities to derive necessary equations from the provided sums and products of the roots. The calculations lead to the conclusion that the last three digits of the desired expression are 560. Various methods, including brute force and systematic substitutions, are discussed to arrive at the solution. The conversation highlights the collaborative nature of problem-solving in mathematics.
#1
anemone
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Let $u,v,x,y$ be complex numbers satisfying
$u+v+x+y=42$,
$uv+ux+uy+vx+vy+xy=2013$,
$u^3+v^3+x^3+y^3+uvx+uvy+uxy+vxy=1337$.
Find the last three digits of $u^4+v^4+x^4+y^4+4uvxy$.
Thanks for participating and your answer is correct.(Clapping)
My method, I believe, is more or less the same as yours.
My solution:
Let $u, v, x, y$ be the roots of a quartic function and $s_n$ represents the sum of the nth powers of those roots. We are asked to evaluate $s_4-uvxy$.
We see that what we have now are
[TABLE="class: grid, width: 700"]
[TR]
[TD]$s_1$[/TD]
[TD]$s_1=42$,[/TD]
[/TR]
[TR]
[TD]$s_2$[/TD]
[TD]$(u+v+x+y)^2=u^2+v^2+x^2+y^2+2(uv+ux+uy+vx+vy+xy)$
By applying the values that we have gotten above into the Newton identities gives the quartic equation $f(a)=a^4-42a^3+2013a^2-45221.75a+\text{product of roots}=a^4-42a^3+2013a^2-45221.75a+uvxy$.
Pardon for my use of the language Englishman. How, I do not know it, I use a "on-line" translator and, already we know " the translations that it realizes ".(Rofl)
Regards.
#5
mathbalarka
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mente oscura said:
Pardon for my use of the language Englishman.
Your English is definitely better than my Spanish.
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A power has two parts. Base and Exponent.
A number 423 in base 10 can be written in other bases as well:
1. 4* 10^2 + 2*10^1 + 3*10^0 = 423
2. 1*7^3 + 1*7^2 + 4*7^1 + 3*7^0 = 1143
3. 7*60^1 + 3*60^0 = 73
All three expressions are equal in quantity. But I have written the multiplier of powers to form numbers in different bases. Is this what place value system is in essence ?