Nuprl Lemma : 33-is-sum-of-three-cubes-implies
This solution to the problem of writing 33 as the sum of three cubes
was found around March 9, 2019 by Andrew Booker using 15 core-years
computation time (over three weeks real time) on a super-computer in Bristol.
The smallest number for which it is unknown whether it is the sum of three
cubes is now 42 (and the next is 114).⋅
∃d,n:ℕ
((∃c:ℤ. ((((2 * d) * ((d * d) + (3 * n * n))) - 33) = (c * c * c) ∈ ℤ))
∨ (∃c:ℤ. (((((2 * d) + 1) * (((d * (d + 1)) + (3 * n * (n + 1))) + 1)) - 33) = (c * c * c) ∈ ℤ)))
Proof
Definitions occuring in Statement :
nat: ℕ,
exists: ∃x:A. B[x],
or: P ∨ Q,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
implies: P ⇒ Q,
false: False,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
exp: i^n,
primrec: primrec(n;b;c),
primtailrec: primtailrec(n;i;b;f),
subtract: n - m
Lemmas referenced :
sum-of-three-cubes-iff,
istype-void,
istype-le,
nat_properties,
decidable__lt,
full-omega-unsat,
intformnot_wf,
intformless_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-less_than,
subtract_wf,
set_subtype_base,
le_wf,
int_subtype_base,
33-is-sum-of-three-cubes,
subtype_base_sq,
decidable__equal_int,
intformeq_wf,
itermMultiply_wf,
itermVar_wf,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_var_lemma,
assert-is_power
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
Error :dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
Error :lambdaFormation_alt,
voidElimination,
hypothesis,
isectElimination,
hypothesisEquality,
productElimination,
setElimination,
rename,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :isect_memberEquality_alt,
Error :universeIsType,
multiplyEquality,
addEquality,
because_Cache,
Error :productIsType,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
intEquality,
Error :inhabitedIsType,
sqequalBase,
equalitySymmetry,
Error :unionIsType,
Error :inlFormation_alt,
instantiate,
cumulativity,
equalityTransitivity,
int_eqEquality,
Error :inrFormation_alt
Latex:
\mexists{}d,n:\mBbbN{}
((\mexists{}c:\mBbbZ{}. ((((2 * d) * ((d * d) + (3 * n * n))) - 33) = (c * c * c)))
\mvee{} (\mexists{}c:\mBbbZ{}. (((((2 * d) + 1) * (((d * (d + 1)) + (3 * n * (n + 1))) + 1)) - 33) = (c * c * c))))
Date html generated:
2019_06_20-PM-02_42_54
Last ObjectModification:
2019_04_11-AM-09_53_09
Theory : num_thy_1
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