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?