Questions asked by George C.

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• `x^4-10x^2+1=0` has one root `x=sqrt(2)+sqrt(3)`. What are the other three roots and why?
  
• The function `f(x) = 1/(1-x)` on `RR` \ `{0, 1}` has the (rather nice) property that `f(f(f(x))) = x`. Is there a simple example of a function `g(x)` such that `g(g(g(g(x)))) = x` but `g(g(x)) != x`?
  
• If you are told that `x^7-3x^5+x^4-4x^2+4x+4 = 0` has at least one repeated root, how might you solve it algebraically?
  
• How can we graph the sawtooth function `x - floor(x)`?
  
• If `z in CC` then what is `sqrt(z^2)`?
  
• Given an integer `n` is there an efficient way to find integers `p, q` such that `abs(p^2-n q^2) <= 1` ?
  
• How do you prove that the set of roots of polynomial equations in one variable with integer coefficients is algebraically closed?
  
• What mathematical conjecture do you know of that is the easiest to explain, but the hardest to attempt a proof of?
  
• How do you show that if `n` is a non-zero integer then `(4/5+3/5i)^n != 1` ?
  
• What simple rigorous ways are there to incorporate infinitesimals into the number system and are they then useful for basic Calculus?
  
• How do you find the zeros of `5x^8+4x^7+20x^5+42x^4+20x^3+4x+5` algebraically?
  
• If the zeros of `x^5+4x+2` are `omega_1`, `omega_2`,.., `omega_5`, then what is `int 1/(x^5+4x+2) dx` ?
  
• What are the zeros of `x^3-8x-4`?
  
• What does cutting squares from an A4 (`297"mm"xx210"mm"`) sheet of paper tell you about `sqrt(2)`?
  
• For what integer values of `k` does `sqrt(x)+sqrt(x+1)=k` have a rational solution?
  
• The sequence `0, 2, 8, 30, 112, 418,...` is defined recursively by `a_0 = 0`, `a_1 = 2`, `a_(n+1) = 4a_n-a_(n-1)`. What is the formula for a general term `a_n`?
  
• If all you know is rational numbers, what is the square root of `2` and how can you do arithmetic with it?
  
• How do you factor completely `f(x) = x^5-2 i x^4-(5+3 i) x^3-(7-3 i) x^2+(6+11 i) x-(1+3 i)` ?
  
• How do you find the discriminant of `x^7-3x^5+x^4-4x^2+4x+4` and what does it tell us?
  
• What are the roots of `8x^3-7x^2-61x+6=0` ?
  
• Why does it make sense to define `sqrt(-1) = i` but not to define `i = sqrt(-1)` ?
  
• Does this word construction (a meditation on Exodus 3) count as poetry, and if so how would you classify it?
  
• Given `f(x) = x^3-3x`, how can you construct an infinitely differentiable one-one function `g(x):RR->RR` with `g(x) = f(x)` in `(-oo, -2] uu [2, oo)`?
  
• Apart from `2, 3` and `3, 5` is there any pair of consecutive Fibonacci numbers which are both prime?
  
• How do you find the zeros of `f(x) = 2x^3+2x^2-2x-1` ?
  
• What are the zeros of `f(x) = x^3-140x^2+7984x-107584`?
  
• A friend chooses two `5` digit positive integers with no common factors and divides one by the other, telling you the result to `12` significant digits. How can you find out what the two numbers were?
  
• If `f(x) = x^5+px+q` (Bring Jerrard normal form) with `p, q` integers, then what are the possible natures of the zeros?
  
• How do you find a `2xx2` matrix `A` with rational coefficients such that `A^2+A+((1,0),(0,1)) = ((0,0),(0,0))` ?
  
• If one of the roots of `x^3-3x+1=0` is given by the rational (companion) matrix `((0,0,-1),(1,0,3),(0,1,0))`, then what rational(?) matrices represent the other two roots?
  
• If `kappa = cos((2pi)/9)+i sin((2pi)/9)` (i.e. the primitive Complex `9`th root of `1`) then how do you express `kappa^8` in the form `a+b kappa + c kappa^2 + d kappa^3 + e kappa^4 + f kappa^5` where `a, b, c, d, e, f` are rational?
  
• Would it be possible to enhance the search facility to find answers within subjects, with particular participants, etc.?
  
• Are octonions numbers?
  
• How can you define a function with domain the whole of `RR` and range the whole of `CC`?
  
• If `a` and `b` are non-zero integers, is it possible for `x^3-ax^2+(a^2-b)x+a(2b-a^2)=0` to have more than one Real root?
  
• What are the roots of `x^3+52x^2+1060x-4624 = 0`?
  
• If `M = ((0,0,0,0,-2),(1,0,0,0,-4),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0))` and `A` is an invertible rational `5xx5` matrix which commutes with `M`, then is `A` necessarily expressible as `A = aM^4+bM^3+cM^2+dM+e` for some scalar factors `a, b, c, d, e`?
  
• How do you prove that `sum_(n=1)^oo (n^(1/n)-1)` diverges?
  
• If `M = ((0,0,-1),(1,0,-1),(0,1,0))` and `A` is an invertible rational `3xx3` matrix which commutes with `M`, then is `A` necessarily expressible as `A = aM^2+bM+cI_3` for some scalar factors `a, b, c`?
  
• Given a function `h` defined by `h(x) = (x-4)/(x+4)`, how do you find a function `f` such that `h = f@f` ?
  
• Is there a rational number `x` such that `sqrt(x)` is irrational, but `sqrt(x)^sqrt(x)` is rational?
  
• What fun, useful, mathematical fact do you know that is not normally taught at school?
  
• Are there equivalent expressions for "eagle-eyed" for senses other than sight?
  
• What possible values can the difference of squares of two Gaussian integers take?
  
• Would it be possible to see how many times an answer has been viewed by other people?
  
• How can we attempt to simplify expressions of the form `sqrt(p+qsqrt(r))` where `p, q, r` are rational?
  
• How can you prove that `1/(0!)+1/(1!)(n-1)+1/(2!)(n-1)(n-2)+1/(3!)(n-1)(n-2)(n-3)+... = 2^(n-1)` ?
  
• What are the zeros of `x^4-26x^2+1` ?
  
• How do you derive exact algebraic formulae for `sin (pi/10)` and `cos (pi/10)` ?
  
• How do you factor `x^5-y^5`?
  
• What are the zeros of `7x^3+70x^2+84x+8=0` ?
  
• Would it be helpful to have some kind of Advanced Algebra subject classification?
  
• Can you find the cube root of a positive integer using a recursively defined sequence?
  
• Which two consecutive integers are such that the smaller added to the square of the larger is `21`?
  
• There does not seem to be a suitable graphic for Algebra level 26 or higher. Should there be?
  
• What is the area of a triangle with vertices `(x_1, y_1)`, `(x_2, y_2)`, `(x_3, y_3)` ?
  
• Would the Lagrange points L4 and L5 of the Sun-Earth system make good locations for a pair of telescopes or does the likely debris around them present a hazard that outweighs the potential advantages?
  
• "Unanswered" filter is cleared on returning to list. Can it be made sticky?
  
• Given a monic cubic function `x^3+bx^2+cx+d` with zeros `alpha, beta, gamma` how can you construct a set of `6xx6` rational matrices that form a field isomorphic to `QQ[alpha, beta, gamma]` ?
  
• How do you simplify `root(3)(135+78sqrt(3))+root(3)(135-78sqrt(3))` ?
  
• What different numbers might be considered conjugates of `1+(root(3)(2))i` and why?