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    <p>It is possible to use Bit-vector arithmetic to solve Diophantine equations. The basic idea is to use <code>ZeroExt</code> to avoid the overflows pointed out by Pad. For example, if we are multiplying two bit-vectors <code>x</code> and <code>y</code> of size <code>n</code>, then we must add <code>n</code> zero bits to <code>x</code> and <code>y</code> to make sure that the result will not overflow. That is, we write:</p> <pre><code> ZeroExt(n, x) * ZeroExt(n, y) </code></pre> <p>The following set of Python functions can be used to "safely" encode any Diophantine equation <code>D(x_1, ..., x_n) = 0</code> into Bit-vector arithmetic. By "safely", I mean if there is a solution that fits in the number of bits used to encode <code>x_1</code>, ..., <code>x_n</code>, then it will eventually be found modulo resources such as memory and time. Using these function, we can encode the Pell's equation <code>x^2 - d*y^2 == 1</code> as <code>eq(mul(x, x), add(val(1), mul(val(d), mul(y, y))))</code>. The function <code>pell(d,n)</code> tries to find values for <code>x</code> and <code>y</code> using <code>n</code> bits.</p> <p>The following link contains the complete example: <a href="http://rise4fun.com/Z3Py/Pehp">http://rise4fun.com/Z3Py/Pehp</a></p> <p>That being said, it is super expensive to solve Pell's equation using Bit-vector arithmetic. The problem is that multiplication is really hard for the bit-vector solver. The complexity of the encoding used by Z3 is quadratic on <code>n</code>. On my machine, I only managed to solve Pell's equations that have small solutions. Examples: <code>d = 982</code>, <code>d = 980</code>, <code>d = 1000</code>, <code>d = 1001</code>. In all cases, I used a <code>n</code> smaller than <code>24</code>. I think there is no hope for equations with very big minimal positive solutions such as <code>d = 991</code> where we need approximately 100 bits to encode the values of <code>x</code> and <code>y</code>. For these cases, I think a specialized solver will perform much better.</p> <p>BTW, the rise4fun website has a small timeout, since it is a shared resource used to run all research prototypes in the Rise group. So, to solve non trivial Pell's equations, we need to run Z3 on our own machines.</p> <pre><code>def mul(x, y): sz1 = x.sort().size() sz2 = y.sort().size() return ZeroExt(sz2, x) * ZeroExt(sz1, y) def add(x, y): sz1 = x.sort().size() sz2 = y.sort().size() rsz = max(sz1, sz2) + 1 return ZeroExt(rsz - sz1, x) + ZeroExt(rsz - sz2, y) def eq(x, y): sz1 = x.sort().size() sz2 = y.sort().size() rsz = max(sz1, sz2) return ZeroExt(rsz - sz1, x) == ZeroExt(rsz - sz2, y) def num_bits(n): assert(n &gt;= 0) r = 0 while n &gt; 0: r = r + 1 n = n / 2 if r == 0: return 1 return r def val(x): return BitVecVal(x, num_bits(x)) def pell(d, n): x = BitVec('x', n) y = BitVec('y', n) solve(eq(mul(x,x), add(val(1), mul(val(d), mul(y, y)))) , x &gt; 0, y &gt; 0) </code></pre>
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