Tue Jan 20 06:36:05 MET 1998 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Number Theory ---------- -- Display the largest 6-digit prime and the smallest 7-digit prime -- => [999983, 1000003] [prevPrime(1000000), nextPrime(1000000)] (1) [999983,1000003] Type: List PositiveInteger Time: 0.02 (IN) + 0.08 (OT) + 0.02 (GC) = 0.12 sec -- Primitive root => 19 191 (2) 191 Type: PositiveInteger Time: 0.02 (OT) = 0.02 sec -- (a + b)^p mod p => a^p + b^p for p prime and a, b in Z_p [Chris Hurlburt] -- See Thomas W. Hungerford, _Algebra_, Springer-Verlag, 1974, p. 121 for a -- more general simplification: (a +- b)^(p^n) => a^(p^n) +- b^(p^n) (a + b)**p :: PrimeField(p) Cannot convert the first argument of PrimeField p to the type PositiveInteger. -- Congruence equations. See Harold M. Stark, _An Introduction to Number -- Theory_, The MIT press, 1984. -- 9 x = 15 mod 21 => x = 4 mod 7 or {4, 11, 18} mod 21 [Stark, p. 68] solve(9*x = 15 :: IntegerMod(21), x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial IntegerMod 21 Variable x -- 7 x = 22 mod 39 => x = 5 mod 13 or 31 mod 39 [Stark, p. 69] solve(7*x = 22 :: IntegerMod(39), x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial IntegerMod 39 Variable x -- x^2 + x + 4 = 0 mod 8 => x = {3, 4} mod 8 [Stark, p. 97] solve(x**2 + x + 4 = 0 :: IntegerMod(8), x) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial IntegerMod 8 Variable x -- x^3 + 2 x^2 + 5 x + 6 = 0 mod 11 => x = 3 mod 11 [Stark, p. 97] solve(x**3 + 2*x**2 + 5*x + 6 = 0 :: PrimeField(11), x) (3) [x= 3] Type: List Equation Fraction Polynomial PrimeField 11 Time: 0.72 (IN) + 0.18 (EV) + 0.18 (OT) = 1.08 sec -- {x = 7 mod 9, x = 13 mod 23, x = 1 mod 2} => x = 151 mod 414 [Stark, -- p. 76] chineseRemainder([7, 13, 1], [9, 23, 2]) (4) 151 Type: PositiveInteger Time: 0.02 (IN) + 0.02 (EV) + 0.03 (OT) + 0.03 (GC) = 0.10 sec -- {5 x + 4 y = 6 mod 7, 3 x - 2 y = 6 mod 7} => x = 1 mod 7, y = 2 mod 7 -- [Stark, p. 76] solve([5*x + 4*y = 6 :: PrimeField(7), 3*x - 2*y = 6 :: PrimeField(7)], [x, y]) (5) [[x= 1,y= 2]] Type: List List Equation Fraction Polynomial PrimeField 7 Time: 1.33 (IN) + 0.08 (EV) + 0.27 (OT) + 0.02 (GC) = 1.70 sec -- 2 x + 3 y = 1 mod 5 => -- (x, y) = {(0, 2), (1, 3), (2, 4), (3, 0), (4, 1)} mod 5 solve(2*x + 3*y = 1 :: PrimeField(5), [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial PrimeField 5 List OrderedVariableList [x,y] -- 2 x + 3 y = 1 mod 6 => [Stark, p. 76] -- (x, y) = {(2, 1), (2, 3), (2, 5), (5, 1), (5, 3), (5, 5)} mod 6 solve(2*x + 3*y = 1 :: IntegerMod(6), [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial IntegerMod 6 List OrderedVariableList [x,y] -- Diophantine equations => x = 2, y = 5 (Wallis) [Stark, p. 147] solve(x**4 + 9 = y**2, [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial Integer List OrderedVariableList [x,y] -- => x = 11, y = 5 (Fermat) [Stark, p. 147] solve(x**2 + 4 = y**3, [x, y]) There are 18 exposed and 3 unexposed library operations named solve having 2 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op solve to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named solve with argument type(s) Equation Polynomial Integer List OrderedVariableList [x,y] -- => (x, y, t, z, w) = (3, 4, 5, 12, 13), (7, 24, 25, 312, 313), ... -- [Stark, p. 154] system:= [x**2 + y**2 = t**2, t**2 + z**2 = w**2] 2 2 2 2 2 2 (6) [y + x = t ,z + t = w ] Type: List Equation Polynomial Integer Time: 0.02 (IN) + 0.02 (EV) + 0.03 (OT) = 0.07 sec solve(system, [x, y, t, z, w]) >> Error detected within library code: system does not have a finite number of solutions initial (7) -> real 58.4 user 14.4 sys 0.5 ------------------------------------------------------------------------------- Fri Jun 13 01:16:45 MET DST 1997 anne % axiom Axiom Computer Algebra System (Release 2.1) Digital Unix on DEC Alpha (AXIOM Sockets) The AXIOM server number is undefined. ----------------------------------------------------------------------------- Issue )copyright to view copyright notices. Issue )summary for a summary of useful system commands. Issue )quit to leave AXIOM and return to shell. ----------------------------------------------------------------------------- initial (1) -> -- ----------[ A x i o m ]---------- -- ---------- Initialization ---------- )set messages autoload off )set messages time on )set quit unprotected -- ---------- Number Theory ---------- -- Rational approximation of sqrt(3) with an error tolerance of 1/500 => 26/15 rationalApproximation(sqrt(3.), 3) 26 (1) -- 15 Type: Fraction Integer Time: 0.02 (IN) + 0.05 (EV) + 0.07 (OT) + 0.07 (GC) = 0.20 sec -- Continued fractions => 3 + 1/(7 + 1/(15 + 1/(1 + 1/(292 + ... continuedFraction(3.1415926535) (2) 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 3 + +---+ + +----+ + +---+ + +-----+ + +---+ + +---+ + +---+ + +---+ | 7 | 15 | 1 | 292 | 1 | 1 | 6 | 2 + 1 | 1 | +----+ + +---+ + ... | 13 | 3 Type: ContinuedFraction Integer Time: 0.03 (EV) + 0.05 (OT) + 0.02 (GC) = 0.10 sec -- => 4 + 1/(1 + 1/(3 + 1/(1 + 1/(8 + 1/(1 + 1/(3 + 1/(1 + 1/(8 + ... -- [Stark, p. 340] continuedFraction(sqrt(23)) There are 2 exposed and 3 unexposed library operations named continuedFraction having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op continuedFraction to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named continuedFraction with argument type(s) AlgebraicNumber continuedFraction(sqrt(23) :: Float) (3) 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 4 + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ | 1 | 3 | 1 | 8 | 1 | 3 | 1 | 8 | 1 + 1 | +---+ + ... | 3 Type: ContinuedFraction Integer Time: 0.13 (IN) + 0.03 (EV) + 0.03 (OT) + 0.03 (GC) = 0.23 sec -- => 1 + 1/(1 + 1/(1 + 1/(1 + ... See Oskar Perron, _Die Lehre von den -- Kettenbr\"uchen_, Chelsea Publishing Company, 1950, p. 52. continuedFraction((1 + sqrt(5))/2 :: Float) (4) 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ + +---+ | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 + 1 | +---+ + ... | 1 Type: ContinuedFraction Integer Time: 0.35 (IN) + 0.02 (EV) + 0.05 (OT) = 0.42 sec -- => 1/(2 x + 1/(6 x + 1/(10 x + 1/(14 x + ... [Perron, p. 353] continuedFraction((exp(1/x) - 1)/(exp(1/x) + 1)) There are 2 exposed and 3 unexposed library operations named continuedFraction having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op continuedFraction to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named continuedFraction with argument type(s) Expression Integer -- => 1/(2 x + 1/(2 x + 1/(2 x + ... (Re x > 0) From Liyang Xu, ``Method -- Derived from Continued Fraction Approximations'', draft. continuedFraction(sqrt(x**2 + 1) - x) There are 2 exposed and 3 unexposed library operations named continuedFraction having 1 argument(s) but none was determined to be applicable. Use HyperDoc Browse, or issue )display op continuedFraction to learn more about the available operations. Perhaps package-calling the operation or using coercions on the arguments will allow you to apply the operation. Cannot find a definition or applicable library operation named continuedFraction with argument type(s) Expression Integer -- ---------- Quit ---------- )quit real 9.3 user 3.0 sys 0.4