Utilizing Hypothesis in the ntheory/polys directory

.gitignore

@@ -87,3 +87,6 @@ sample.tex
 
 # Files for pyodide test
 pyodide
+
+# Ignore hypothesis files
+.hypothesis

sympy/ntheory/tests/test_tau_hypothesis.py

@@ -0,0 +1,18 @@
+from hypothesis import given
+from hypothesis import strategies as st
+from sympy import divisors
+from sympy.ntheory.primetest import is_square
+from sympy.ntheory import totient
+from sympy.ntheory import divisor_sigma
+
+
+@given(n=st.integers(1, 10**10))
+def test_tau_hypothesis(n):
+    div = divisors(n)
+    tau_n = len(div)
+    if tau_n % 2 == 1:  # tau(n) odd iff n is perfect square
+        assert is_square(n)
+    sigmas = [divisor_sigma(i) for i in div]
+    totients = [totient(n // i) for i in div]
+    mul = [a * b for a, b in zip(sigmas, totients)]
+    assert n * tau_n == sum(mul)

sympy/ntheory/tests/test_totient_hypothesis.py

@@ -0,0 +1,15 @@
+from hypothesis import given
+from hypothesis import strategies as st
+from sympy.ntheory import totient
+from sympy import divisors
+import random
+
+
+@given(n=st.integers(1,10**10))
+def test_totient_hypothesis(n):
+    assert totient(n) <= n
+    div = divisors(n)
+    totients = [totient(i) for i in div]
+    assert n == sum(totients)
+    d = random.choice(div)
+    assert totient(n // (n // d)) == totient(d)

sympy/polys/tests/test_gcd_hypothesis.py

@@ -0,0 +1,27 @@
+from hypothesis import given
+from hypothesis import strategies as st
+from sympy.abc import x
+from sympy.polys.polytools import Poly
+
+
+@given(
+    coefficients1=st.lists(st.integers()),
+    coefficients2=st.lists(st.integers()),
+    coefficients3=st.lists(st.integers()),
+)
+def test_gcd_hypothesis(coefficients1, coefficients2, coefficients3):
+    # instantiate polynomials
+    f = Poly(coefficients1, x, domain="ZZ")
+    g = Poly(coefficients2, x, domain="ZZ")
+    r = Poly(coefficients3, x, domain="ZZ")
+
+    # get gdc
+    gcd_1 = f.gcd(g)
+    gcd_2 = g.gcd(f)
+
+    assert gcd_1 == gcd_2
+
+    # multiply by r
+    gcd_3 = g.gcd(f + r * g)
+
+    assert gcd_1 == gcd_3

sympy/polys/tests/test_poly_hypothesis.py

@@ -0,0 +1,23 @@
+from hypothesis import given
+from hypothesis import assume
+from hypothesis import strategies as st
+from sympy.abc import x
+from sympy.polys.polytools import Poly
+
+
+@given(coefficients1=st.lists(st.integers()), coefficients2=st.lists(st.integers()))
+def test_poly_hypothesis(coefficients1, coefficients2):
+    # non-zero division check.
+    assume(len(coefficients2) * [0] != coefficients2)
+
+    # Integer case
+    f_z = Poly(coefficients1, x, domain="ZZ")
+    g_z = Poly(coefficients2, x, domain="ZZ")
+    remainder_z = f_z.rem(g_z)
+    assert g_z.degree() > remainder_z.degree() or remainder_z.degree() == 0
+
+    # Rational case
+    f_q = Poly(coefficients1, x, domain="QQ")
+    g_q = Poly(coefficients2, x, domain="QQ")
+    remainder_q = f_q.rem(g_q)
+    assert g_q.degree() > remainder_q.degree() or remainder_q.degree() == 0