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