demo-parity-05 true

category parity

Summary

Given

\(n, m \in \mathbb{Z},\quad n \in 2\mathbb{Z} + 1,\quad m \in 2\mathbb{Z} + 1\)

Claim

\(\displaystyle \left(-1\right)^{n \, m} = -1\)

Expected

true equal at every admissible point

the two sides are equal at every admissible point

Answers (latest run per system; open code for the reproducer that ran)

system outcome answer steps (from the trace) s run query (as sent) detail

none

always-none no verdict null 0.0 2026-10-13 17:20:05 df65d778fc5c (integer n) ∧ (integer m) ∧ (odd n) ∧ (odd m) ⊢ (= (pow -1 (* n m)) -1) null

sympy

sympy-ask no verdict null difference → (-1)**(m*n) + 1 · ask → None 0.071 2026-10-13 17:20:46 df65d778fc5c Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
Assum = And(Q.integer(n), Q.integer(m), Q.odd(n), Q.odd(m))
Residual = Pow(-1, n * m) - (-1)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`(-1)*(mn) + 1`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-ask no verdict null difference → (-1)**(m*n) + 1 · ask → None 0.03 2026-10-12 20:21:28 d7af4def9aaa Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
Assum = And(Q.integer(n), Q.integer(m), Q.odd(n), Q.odd(m))
Residual = Pow(-1, n * m) - (-1)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`(-1)*(mn) + 1`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

sympy-refine solved true difference → (-1)**(m*n) + 1 · refine → 0 0.051 2026-10-13 17:20:49 df65d778fc5c Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
Assum = And(Q.integer(n), Q.integer(m), Q.odd(n), Q.odd(m))
Residual = Pow(-1, n * m) - (-1)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`(-1)*(mn) + 1`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

sympy-refine solved true difference → (-1)**(m*n) + 1 · refine → 0 0.041 2026-10-12 20:21:30 d7af4def9aaa Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
Assum = And(Q.integer(n), Q.integer(m), Q.odd(n), Q.odd(m))
Residual = Pow(-1, n * m) - (-1)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`(-1)*(mn) + 1`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.008 2026-10-13 17:20:51 df65d778fc5c Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True, odd=True)
m = Symbol('m', integer=True, odd=True)
Residual = Pow(-1, n * m) - (-1)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.008 2026-10-12 20:21:32 d7af4def9aaa Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True, odd=True)
m = Symbol('m', integer=True, odd=True)
Residual = Pow(-1, n * m) - (-1)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

wolfram

wolfram-fullsimplify solved true difference → 1 + (-1)^(m*n) · result → 0 0.022 2026-10-13 17:20:59 df65d778fc5c Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → 1 + (-1)^(m*n) · result → 0 0.022 2026-10-13 17:20:56 df65d778fc5c Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → 1 + (-1)^(m*n) · result → 0 0.022 2026-10-13 17:20:52 df65d778fc5c Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

Older runs (11)

system outcome answer steps (from the trace) s run query (as sent) detail

wolfram-fullsimplify solved true difference → 1 + (-1)^(m*n) · result → 0 0.023 2026-10-13 16:39:09 03f5149718d0 Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → 1 + (-1)^(m*n) · result → 0 0.022 2026-10-13 16:39:06 03f5149718d0 Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → 1 + (-1)^(m*n) · result → 0 0.023 2026-10-13 16:39:02 03f5149718d0 Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

sympy-simplify solved true difference → 0 · simplify → 0 0.007 2026-10-13 16:39:01 03f5149718d0 Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = simplify(sin(W)**2 + cos(W)**2)
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True, odd=True)
m = Symbol('m', integer=True, odd=True)
Residual = Pow(-1, n * m) - (-1)
Result = simplify(Residual)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`0`
simplify	`simplify(Residual)`	→	`0`
answer			`true`

sympy-refine solved true difference → (-1)**(m*n) + 1 · refine → 0 0.051 2026-10-13 16:38:59 03f5149718d0 Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) true

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = refine(Abs(W), Q.positive(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
Assum = And(Q.integer(n), Q.integer(m), Q.odd(n), Q.odd(m))
Residual = Pow(-1, n * m) - (-1)
Result = refine(Residual, Assum)
# true iff the result is 0; false only if it is a definite nonzero number
Answer = True if Result == 0 else False if (Result.is_number and Result.is_finite and Result.is_zero is False) else None
print(Result, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`(-1)*(mn) + 1`
refine	`refine(Residual, Assum)`	→	`0`
answer			`true`

always-none no verdict null 0.0 2026-10-13 16:38:56 03f5149718d0 (integer n) ∧ (integer m) ∧ (odd n) ∧ (odd m) ⊢ (= (pow -1 (* n m)) -1) null

sympy-ask no verdict null difference → (-1)**(m*n) + 1 · ask → None 0.071 2026-10-13 16:38:56 03f5149718d0 Q.integer(n) ∧ Q.integer(m) ∧ Q.odd(n) ∧ Q.odd(m) ⊢ Eq(Pow((-1), (n*m)), (-1)) null

code

setup — not timed

from sympy import *
import sympy

SympyVersion = sympy.__version__
# warm-up: a fixed generic expression, never case content
W = Symbol('W')
Warmup = ask(Q.positive(W**2 + 1), Q.real(W))
# every query then runs in its own forked process, starting from this state:
# the expression caches are dropped (imported and lazily loaded modules stay)
sympy.core.cache.clear_cache()

n = Symbol('n', integer=True)
m = Symbol('m', integer=True)
Assum = And(Q.integer(n), Q.integer(m), Q.odd(n), Q.odd(m))
Residual = Pow(-1, n * m) - (-1)
# not Q.eq: in SymPy 1.14 ask(Q.eq(x**2, x), Q.real(x)) is wrongly False
Answer = ask(Q.zero(Residual), Assum)
print(Residual, Answer, sep='\n')

difference	`Pow(-1, n * m) - (-1)`	→	`(-1)*(mn) + 1`
ask	`Q.zero(Residual)`	→	`None`
answer			`null`

wolfram-fullsimplify solved true difference → 1 + (-1)^(m*n) · result → 0 0.023 2026-10-12 20:21:41 d7af4def9aaa Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = FullSimplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = FullSimplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`FullSimplify[Residual, Assum]`	→	`0`
answer			`true`

wolfram-refine solved true difference → 1 + (-1)^(m*n) · result → 0 0.022 2026-10-12 20:21:37 d7af4def9aaa Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Refine[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = Refine[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`Refine[Residual, Assum]`	→	`0`
answer			`true`

wolfram-simplify solved true difference → 1 + (-1)^(m*n) · result → 0 0.022 2026-10-12 20:21:34 d7af4def9aaa Element[n, Integers] ∧ Element[m, Integers] ∧ Element[((n) - 1)/2, Integers] ∧ Element[((m) - 1)/2, Integers] ⊢ (Power[(-1), (n*m)] == (-1)) true

code

setup — not timed

(
  $HistoryLength = 0;
  $Messages = {};
  Warmup = Simplify[Sin[W]^2 + Cos[W]^2, Element[W, Reals]];
  GenericWarmup = Quiet[{Simplify[Abs[W] + W, W < 0], Simplify[Sqrt[W^2] - W, W > 0], Refine[Sqrt[W^2], W > 0], Simplify[Log[W V] - Log[W] - Log[V], W > 0 && V > 0], Simplify[Log[-W] - Log[W] - I Pi, W < 0], Simplify[(-1)^(2 W) - 1, Element[W, Integers]], Simplify[Mod[W + 1, 2], Element[W, Integers] && Element[(W - 1)/2, Integers]], Simplify[ArcTan[W] + ArcTan[1/W] - Pi/2, W > 0], FullSimplify[ArcCot[W] - ArcTan[1/W], W > 0], Simplify[Abs[Exp[I W]] - 1, Element[W, Reals]], Simplify[Re[W] - (W + Conjugate[W])/2], Simplify[Gamma[W + 1] - W!, Element[W, Integers] && W > 0], Simplify[Floor[W] - W, Element[W, Integers]], Simplify[Sqrt[W] Sqrt[V] - Sqrt[W V], W > 0 && V > 0], Simplify[HeavisideTheta[W] - 1, W > 0], Simplify[2^W 2^V - 2^(W + V), Element[W, Reals] && Element[V, Reals]], Simplify[Sum[V, {V, 1, W}] - W (W + 1)/2, Element[W, Integers] && W > 0]}];
  Quiet[Check[Message[General::argx, "W", 0], Null]];
  {$Version, Warmup, Length[GenericWarmup]}
)

(* evaluated in a forked copy of the setup kernel: TimeConstrained would not fire here (no timer thread), so the parent enforces a hard timeout of 9 s by killing this process *)
(
  Assum = (Element[n, Integers]) && (Element[m, Integers]) && (Element[((n) - 1)/2, Integers]) && (Element[((m) - 1)/2, Integers]);
  Residual = ((-1)^(n*m)) - (-1);
  Result = Simplify[Residual, Assum];
  Answer = Which[Result === 0, True, NumericQ[Result] && TrueQ[Result != 0], False, True, Null];
  {Residual, Result, Answer}
)

difference	`((-1)^(n*m)) - (-1)`	→	`1 + (-1)^(m*n)`
result	`Simplify[Residual, Assum]`	→	`0`
answer			`true`

always-none no verdict null 0.0 2026-10-12 20:21:33 d7af4def9aaa (integer n) ∧ (integer m) ∧ (odd n) ∧ (odd m) ⊢ (= (pow -1 (* n m)) -1) null

Source (parity.sexp)

(case demo-parity-05 (category parity)
  (var n integer) (var m integer)
  (assume (odd n)) (assume (odd m))
  (ask (= (pow -1 (* n m)) -1))
  (expect true))