Integer and polynomial arithmetic

You may force integer arithmetic by setting flag 10 (shortcut $\alpha$-right shift-CST or 10 SF). Otherwise, polynomial arithmetic is assumed. This is important for instructions like GCD3 or ABCUV.

• DIV2: euclidean division. Stack 2 is the quotient, stack 1 the remainder.
• GCD1:
  returns the greatest common divisor d of two objects a and b (integers, Gauß integers, polynomials). Examples:
  'X^2+2*X+1' 'X^2+3*X+2' GCD1 returns 'X+1'
25 15 GCD1 returns 5
  If flag 12 is clear returns 1.
• LCM1: lowest common multiple (GCD1 $(a,b)\times$LCM1 $(a,b)=a \times b$). Examples:
  'X^2+2*X+1' 'X^2+3*X+2' LCM1 returns '(X^2+2*X+1)*(X+2)'
25 15 LCM1 returns 75
• GCD3: extended gcd algorithm, given x and y returns d , u and v s.t.:
  
  ux+vy=d
  
  
  (d is a multiple of the gcd of x and y by an invertible, i.e. an integer in the univariate case)
• ABCUV: (Bezout identity)
  solve the equation c=ax+by Examples:
  

  'X^2+2*X+1' 'X^2+3*X+2' 'X+1' -> -1 1 1
  'X^2+2*X+1' 'X^2+3*X+2' 1 -> 0
  

  This means for the first case that:
  
  (X+1) = (X²+2X+1)*(-1) + (X²+3X+2)*1
  
  
  as in the second case there is no solution because the gcd of x²+x+1 and x²+3x+2 does not divide 1.

• LGCD: returns the gcd of a list of objects.

• SIMP2: simplifies two objects by dividing them by their GCD. Sets flags 12, 14 and 15. Ex:
  stk2: 9, stk1: 6 SIMP2 -> stk2: 3, stk1: 2

• DIVIS: gives a list of divisors of an object. Example:
  21 -> { 1 7 3 21 }

• fact and comb: like the built-in FACT and COMB instructions but for long integers.
• EULER: Euler indicatrix
  Given an integer n, returns an integer e: the number of integers lower than and prime with n.
  Example:
  for n=25, e=20 because 1, 2, 3, 4, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 22, 23, 24 are prime with 25.

• PA2B2 (kernel library):
  Given a prime p which is 1 modulo 4, returns a complex z=a+ib such that |z|²=a²+b²=p. Used for factorization of Gauß integers.

• XFRC:
  Same as $\rightarrow$Q but handles quadratic irrationals (recognize a quadratic irrational if its expansion in continued fraction is ultimately periodic of period less or equal to 3) Examples:
  
  
  $$1.20710678118 \rightarrow \frac{1}{2} +\frac{1}{2} \sqrt{2} \quad 1.5 \rightarrow \frac{3}{2}$$
  
  

• ORND:
  round object at stack level 2, stack level 1 is the expected denominator of all rationals of the object. For example, .49999999999+.2000000001*x 10 ORND returns .5+.2*x.