Taxicab.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

7 declarations, 2 contain sorry, 0 declared as axioms

Taxicab numbers

A taxicab number for natural numbers $k, m, n$ is the smallest number $x$ that can be expressed as a sum of $m$ positive $k$-th powers in at least $n$ distinct ways. The most famous taxicab number is $ 1729 = 1³ + 12³ = 9³ + 10³, $ also known as the Hardy–Ramanujan number.

However, a taxicab number is not known for $k=5$, $m=2$, and any $n ≥ 2$: No positive integer is known that can be written as the sum of two 5th powers in more than one way, and it is not known whether such a number exists.

In particular, it is not known whether there exists a taxicab number for $k=5$, $m=2$, and $n=2$.

References:

Imports (1)

FormalConjecturesUtil

def Taxicab.IsTaxicabFor' Documented declaration: IsTaxicabFor' line 55 source
Source documentation
$x$ is a candidate for being a taxicab number for $k, m, n$ if there exist at least $n$ distinct multisets of $m$ positive integers such that the sum of the $k$-th powers of the elements of each multiset is $x$. Using multisets means that two representations that differ only in the order of their terms count as one way.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: Exists, Finset, Multiset, And, LE.le, Finset.card, Membership.mem, Eq, Multiset.card, Not, OfNat.ofNat, Multiset.sum, Multiset.map, HPow.hPow
10 instances, coercions and auxiliary constants left out.
```
def IsTaxicabFor' (k m n x : ℕ) : Prop :=
…
```
theorem Taxicab.taxicab_1729 $1729$ is a possible taxicab number for $k=3, m=2, n=2$ line 62 source
Source documentation
$1729$ is a possible taxicab number for $k=3, m=2, n=2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Taxicab.IsTaxicabFor', OfNat.ofNat, Nat
Proof uses: Exists.intro, Finset, Multiset, And, LE.le, Finset.card, Membership.mem, Eq, Multiset.card, Not, Multiset.sum, Multiset.map, HPow.hPow, Insert.insert, Singleton.singleton, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true and 1 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem taxicab_1729 : IsTaxicabFor' 3 2 2 1729 := by
…
```
Mentions: Taxicab.IsTaxicabFor'
def Taxicab.IsTaxicabFor $x$ is a taxicab number if it is the smallest number that can be expressed as a sum of $m$ positive $k$-th powers in at least $n$ distinct ways line 70 source
Source documentation
$x$ is a taxicab number if it is the smallest number that can be expressed as a sum of $m$ positive $k$-th powers in at least $n$ distinct ways.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat
Proof uses: IsLeast, Set.ofPred, Taxicab.IsTaxicabFor'
1 instances, coercions and auxiliary constants left out.
```
def IsTaxicabFor (k m n : ℕ) (x : ℕ) : Prop :=
…
```
theorem Taxicab.taxicab_4' line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Taxicab.IsTaxicabFor', OfNat.ofNat, Nat
Proof uses: Exists.intro, Finset, Multiset, And, LE.le, Finset.card, Membership.mem, Eq, Multiset.card, Not, Multiset.sum, Multiset.map, HPow.hPow, Insert.insert, Singleton.singleton, of_decide_eq_true, id, Bool, Decidable.decide, Bool.true and 1 more
22 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
theorem Taxicab.taxicab_4 $4$ is the taxicab number for $k=1, m=2, n=2$ line 80 source
Source documentation
$4$ is the taxicab number for $k=1, m=2, n=2$.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Taxicab.IsTaxicabFor, OfNat.ofNat, Nat
Proof uses: And.intro, Membership.mem, Set, Set.ofPred, Taxicab.IsTaxicabFor', lowerBounds, Taxicab.taxicab_4', Finset, Multiset, And, LE.le, Finset.card, Eq, Multiset.card, Not, Multiset.sum, Multiset.map, HPow.hPow, Exists, Ne and 35 more
97 instances, coercions and auxiliary constants left out.

Tactic trace: 28 steps
Open to load the checked before-and-after goal states.
```
theorem taxicab_4 : IsTaxicabFor 1 2 2 4 := by
…
```
Mentions: Taxicab.IsTaxicabFor
theorem Taxicab.taxicab_for_5_2_2 Taxicab number for $k=5$, $m=2$, and $n=2$ is not known sorry line 101 source
Source documentation
Taxicab number for $k=5$, $m=2$, and $n=2$ is not known. Whether such a number exists is also not known.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, Taxicab.IsTaxicabFor, OfNat.ofNat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
2 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem taxicab_for_5_2_2 : answer(sorry) ↔ ∃ x : ℕ, IsTaxicabFor 5 2 2 x := by
…
```
Mentions: Taxicab.IsTaxicabFor
theorem Taxicab.taxicab_for_5_2_n Taxicab number for $k=5$ and $m=2$ is not-known for any $n ≥ 2$ sorry line 107 source
Source documentation
Taxicab number for $k=5$ and $m=2$ is not-known for any $n ≥ 2$. Whether such a number exists is also not known.

depends on sorry what this means
Statement uses: Iff, True, Exists, Nat, And, GE.ge, OfNat.ofNat, Taxicab.IsTaxicabFor
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem taxicab_for_5_2_n : answer(sorry) ↔ ∃ n : ℕ, n ≥ 2 ∧ (∃ x : ℕ, IsTaxicabFor 5 2 n x)
  := by sorry
…
```
Mentions: Taxicab.IsTaxicabFor