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NumberTheory

  

InhomogeneousDiophantine

  

inhomogeneous Diophantine approximation

 

Calling Sequence

Parameters

Description

Examples

Compatibility

Calling Sequence

InhomogeneousDiophantine(ineqs, xvars, yvars)

InhomogeneousDiophantine(cfs, alpha, real_errors)

InhomogeneousDiophantine(cfs, alpha, adicities, padic_errors)

Parameters

ineqs

-

inequality or set of inequalities with abs or valuep

xvars

-

name or set of names

yvars

-

name or set of names

cfs

-

convertible to a Matrix of real numbers

alpha

-

convertible to a Vector of real numbers

adicities

-

convertible to a Vector of prime numbers

real_errors

-

convertible to a Vector of real numbers

padic_errors

-

convertible to a Vector of positive integers

Description

• 

The InhomogeneousDiophantine function finds a solution x1,,xn,y1,,ym over the integers to a set of inequalities of the form

a1,1x1+a1,nxn+...α1y1err1

...

am,1x1+am,nxn+...αmymerrm

  

or

padic:−valuepa1,1x1+a1,nxn+...α1y1,p1p1err1

...

padic:−valuepam,1x1+am,nxn+...αmym,pmpmerrm

  

where padic:−valuep is the p-adic valuation.

• 

The inequalities can be described explicitly, corresponding to the first calling sequence, or implicitly, corresponding to the other calling sequences.

• 

If the first calling sequence is used, then the return value is of the form

x1=s1,...,xn=sn,y1=t1,...,ym=tm

• 

If the other calling sequences are used, then the return value is a two-element list corresponding to the x values and the y values,

s1,...,sn,t1,...,tm

Examples

withNumberTheory:

withpadic:

InhomogeneousDiophantine0.01log2x+24log5y8312zⅇ2.5u107,3.7ⅇ2x+y+313z513v103,x,y,z,u,v

x=−5327,y=2703,z=−1639,v=145975,u=127069

(1)

An equivalent Matrix form calling sequence is:

InhomogeneousDiophantine0.01log2,24log5,8312,3.7ⅇ2,1,313,ⅇ2.5,513,107,103

6333,−8617,3579,−382405,−176598

(2)

The solutions may be different but both are valid.

InhomogeneousDiophantinevalueplog5x+log7ylog5u,3320,valuep1xlog7+log11ylog7v,5515,valueplog3x+ⅇ7ylog3w,7712,x,y,u,v,w

x=−15516275,y=6404775,w=−9747866955,u=−1192024656,v=−27148890349

(3)

The error list for the p-adic cases are negatives of the exponents on the adicities.

InhomogeneousDiophantinelog5,log7,1log7,log11,log3,ⅇ7,log5,log7,log3,3,5,7,20,15,12

−328700,−11704900,−1192426818,−27149007027,−9747320216

(4)

Compatibility

• 

The NumberTheory[InhomogeneousDiophantine] command was introduced in Maple 2016.

• 

For more information on Maple 2016 changes, see Updates in Maple 2016.

See Also

isolve

NumberTheory

NumberTheory[HomogeneousDiophantine]

padic[valuep]