avewright/proofwiki-math huggingface dataset:
Theorem: Arithmetic Sequence of 4 Terms with 3 Distinct Prime Factors

The arithmetic sequence:
:$30, 66, 102, 138$
is the smallest of $4$ terms which consists entirely of positive integers each with $3$ distinct prime factors.

Proof:
We demonstrate that this is indeed an arithmetic sequence:
{{begin-eqn}}
{{eqn | l = 66 - 30
      | r = 36
}}
{{eqn | l = 102 - 66
      | r = 36
}}
{{eqn | l = 138 - 102
      | r = 36
}}
{{end-eqn}}
demonstrating the common difference of $36$.
Then we note:
{{begin-eqn}}
{{eqn | l = 30
      | r = 2 \times 3 \times 5
}}
{{eqn | l = 66
      | r = 2 \times 3 \times 11
}}
{{eqn | l = 102
      | r = 2 \times 3 \times 17
}}
{{eqn | l = 138
      | r = 2 \times 3 \times 23
}}
{{end-eqn}}
It is understood that '''smallest arithmetic sequence''' means:
:integer sequence of length $4$ whose terms are in arithmetic sequence such that their sum is the smallest of all such sequences.
To demonstrate that this sequence is indeed the smallest, according to this definition, we list all positive integers less than $138$, each with $3$ distinct prime factors.
They are:
{{begin-eqn}}
{{eqn | l = 30
      | r = 2 \times 3 \times 5
}}
{{eqn | l = 42
      | r = 2 \times 3 \times 7
}}
{{eqn | l = 66
      | r = 2 \times 3 \times 11
}}
{{eqn | l = 70
      | r = 2 \times 5 \times 7
}}
{{eqn | l = 78
      | r = 2 \times 3 \times 13
}}
{{eqn | l = 102
      | r = 2 \times 3 \times 17
}}
{{eqn | l = 105
      | r = 3 \times 5 \times 7
}}
{{eqn | l = 110
      | r = 2 \times 5 \times 11
}}
{{eqn | l = 114
      | r = 2 \times 3 \times 17
}}
{{eqn | l = 130
      | r = 2 \times 5 \times 13
}}
{{end-eqn}}
One cannot pick any other $4$ numbers from the list to form an arithmetic sequence.
Hence any further such arithmetic sequence will contain larger terms whose sum will be consequently larger.
{{qed}}
Theorem: Sum of Successive Powers in 2 ways

$31$ and $8191$ can be expressed as the sum of successive powers starting from $1$ in in $2$ different ways.

Proof:
{{begin-eqn}}
{{eqn | l = 31
      | r = 1 + 5 + 5^2
      | c = 
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + 2^4
      | c = 
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 8191
      | r = 1 + 90 + 90^2
      | c = 
}}
{{eqn | r = 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8 + 2^9 + 2^{10}+ 2^{11}+ 2^{12}
      | c = 
}}
{{end-eqn}}
These are the only two examples known.
{{qed}}
Theorem: 31 is Smallest Prime whose Reciprocal has Odd Period

$31$ is the smallest prime number to have a decimal expansion of the reciprocal with an odd period greater than $1$:
:$\dfrac 1 {31} = 0 \cdotp \dot 03225 \, 80645 \, 1612 \dot 9$

Proof:
From Reciprocal of $31$:
{{:Reciprocal of 31}}
Counting the digits, it is seen that this has a period of recurrence of $15$, an odd integer.
The prime numbers less than $31$ are $2$, $3$, $5$, $7$, $11$, $13$, $17$, $19$, $23$, $29$.
We investigate the reciprocal of each of these:
{{begin-eqn}}
{{eqn | l = \dfrac 1 2
      | r = 0 \cdotp 5
      | c = Reciprocal of $2$: not recurring
}}
{{eqn | l = \dfrac 1 3
      | r = 0 \cdotp \dot 3
      | c = Reciprocal of $3$: period $1$
}}
{{eqn | l = \dfrac 1 5
      | r = 0 \cdotp 2
      | c = Reciprocal of $5$: not recurring
}}
{{eqn | l = \dfrac 1 7
      | r = 0 \cdotp \dot 14285 \dot 7
      | c = Reciprocal of $7$: period $6$
}}
{{eqn | l = \dfrac 1 {11}
      | r = 0 \cdotp \dot 0 \dot 9
      | c = Reciprocal of $11$: period $2$
}}
{{eqn | l = \dfrac 1 {13}
      | r = 0 \cdotp \dot 07692 \dot 3
      | c = Reciprocal of $13$: period $6$
}}
{{eqn | l = \dfrac 1 {17}
      | r = 0 \cdotp \dot 05882 \, 35294 \, 11764 \dot 7
      | c = Reciprocal of $17$: period $16$
}}
{{eqn | l = \dfrac 1 {19}
      | r = 0 \cdotp \dot 05263 \, 15789 \, 47368 \, 42 \dot 1
      | c = Reciprocal of $19$: period $18$
}}
{{eqn | l = \dfrac 1 {23}
      | r = 0 \cdotp \dot 04347 \, 82608 \, 69565 \, 21739 \, 1 \dot 3
      | c = Reciprocal of $23$: period $22$
}}
{{eqn | l = \dfrac 1 {29}
      | r = 0 \cdotp \dot 03448 \, 27586 \, 20689 \, 65517 \, 24137 \, 93 \dot 1
      | c = Reciprocal of $29$: period $28$
}}
{{end-eqn}}
and it is seen that none has an odd period greater than $1$.
Hence the result.
{{qed}}
Theorem: Properties of Periodic Part of Reciprocal of 31

We have from Reciprocal of $31$ that the decimal expansion of the reciprocal of $31$ is:
{{:Reciprocal of 31}}
Then:
{{begin-eqn}}
{{eqn | l = 032258 \times 2
      | r = 64 \, 516
}}
{{eqn | l = 032258 \times 4
      | r = 129 \, 032
}}
{{eqn | l = 032258 \times 5
      | r = 161 \, 290
}}
{{eqn | l = 032258 \times 7
      | r = 225 \, 806
}}
{{eqn | l = 032258 \times 8
      | r = 258 \, 064
}}
{{eqn | l = 032258 \times 9
      | r = 290 \, 322
}}
{{eqn | l = 032258 \times 14
      | r = 451 \, 612
}}
{{eqn | l = 032258 \times 16
      | r = 516 \, 128
}}
{{eqn | l = 032258 \times 18
      | r = 580 \, 644
}}
{{eqn | l = 032258 \times 19
      | r = 612 \, 902
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 03225 + 80645 + 16129
      | r = 99 \, 999
}}
{{eqn | l = 032 + 258 + 065 + 416 + 129
      | r = 900
}}
{{end-eqn}}

Proof:
Verified by calculation.
Theorem: Smallest Adjacent Happy Numbers

The smallest adjacent happy numbers are $31$ and $32$.

Proof:
This can be determined by testing all the positive integers in succession for happiness.
Checking $31$ and $32$:
{{begin-eqn}}
{{eqn | l = 31
      | o = \to
      | r = 3^2 + 1^2 = 9 + 1 = 10
      | c = 
}}
{{eqn | o = \to
      | r = 1^2 = 1
      | c = and so $31$ is happy.
}}
{{end-eqn}}
{{begin-eqn}}
{{eqn | l = 32
      | o = \to
      | r = 3^2 + 2^2 = 9 + 4 = 13
      | c = 
}}
{{eqn | o = \to
      | r = 1^2 + 3^2 = 1 + 9 = 10
}}
{{eqn | o = \to
      | r = 1^2 = 1
      | c = and so $32$ is happy.
}}
{{end-eqn}}
{{qed}}
Theorem: Increasing Mapping Preserves Lower Bounds

Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets.
Let $f:S \to S'$ be an increasing mapping.
Let $x \in S$, $X \subseteq S$ such that
:$x$ is lower bound for $X$.
Then $f \left({x}\right)$ is lower bound for $f \left[{X}\right]$.

Proof:
Let $y \in f\left[{X}\right]$.
By definition of image of set:
:$\exists z \in X: y = f \left({z}\right)$
By definition of lower bound:
:$x \preceq z$
Thus by definition of increasing mapping:
:$f \left({x}\right) \preceq' y$
{{qed}}
Theorem: Order Isomorphism Preserves Lower Bounds

Let $L = \struct {S, \preceq}$, $L' = \struct {S', \preceq'}$ be ordered sets.
Let $f: S \to S'$ be an order isomorphism between $L$ and $L'$.
Let $x \in S$, $X \subseteq S$.
Then $x$ is lower bound for $X$ {{iff}} $\map f x$ is lower bound for $f \sqbrk X$.

Proof:
By definition of order isomorphism:
:$f$ is an order embedding.
Theorem: Order Embedding is Increasing Mapping

Let $\left({S_1, \preceq_1}\right)$, $\left({S_2, \preceq_2}\right)$ be ordered sets.
Let $f:S_1 \to S_2$ be an order embedding.
Then $f$ is increasing mapping.

Proof:
By definition of order embedding:
:$\forall x, y \in S_1: x \preceq_1 y \implies f\left({x}\right) \preceq_2 f\left({y}\right)$
Hence $f$ is an increasing mapping.
{{qed}}
Category:Order Embeddings
Category:Increasing Mappings
2dys8l0sul24o8x0jrg12yha84gkk4f
Theorem: Order Isomorphism Preserves Upper Bounds

Let $L = \struct {S, \preceq}$, $L' = \struct {S', \preceq'}$ be ordered sets.
Let $f: S \to S'$ be an order isomorphism between $L$ and $L'$.
Let $x \in S$, $X \subseteq S$.
Then:
:$x$ is an upper bound for $X$
{{iff}}:
:$\map f x$ is an upper bound for $f \sqbrk X$.

Proof:
By definition of order isomorphism:
:$f$ is an order embedding.
Theorem: Increasing Mapping Preserves Upper Bounds

Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets.
Let $f:S \to S'$ be an increasing mapping.
Let $x \in S$, $X \subseteq S$ such that
:$x$ is upper bound for $X$.
Then $f\left({x}\right)$ is upper bound for $f\left[{X}\right]$.

Proof:
Let $y \in f\left[{X}\right]$.
By definition of image of set:
:$\exists z \in X: y = f\left({z}\right)$
By definition of upper bound:
:$z \preceq x$
Thus by definition of increasing mapping:
:$y \preceq' f\left({x}\right)$
{{qed}}
Theorem: Order Isomorphism Preserves Infima and Suprema

Let $L = \struct {S, \preceq}$ and $L' = \struct {S', \preceq'}$ be ordered sets.
Let $f: S \to S'$ be an order isomorphism between $L$ and $L'$.
Then $f$ preserves infima and suprema.

Proof:
=== $f$ preserves infima ===
Let $X$ be a subset of $S$ such that
:$X$ admits an infimum in $L$.
By definition of infimum:
:$\inf X$ is lower bound for $X$.
Thus by Order Isomorphism Preserves Lower Bounds:
:$\map f {\inf X}$ is lower bound for $f \sqbrk X$.
We will prove that
:$\forall x \in S': x$ is lower bound for $f \sqbrk X \implies x \preceq' \map f {\inf X}$
Let $x \in S'$ such that
:$x$ is lower bound for $f \sqbrk X$.
By definitions of order isomorphism and bijection:
:$f$ is a surjection.
By definition of surjection:
:$S' = f \sqbrk X$
By definition of image of set:
:$\exists y \in S: x = \map f y$
By Order Isomorphism Preserves Lower Bounds:
:$y$ is lower bound for $X$.
By definition of infimum:
:$y \preceq \inf X$
Thus by definition of order isomorphism:
:$x \preceq' \map f {\inf X}$
{{qed|lemma}}
Thus by definition of infimum:
:$f \sqbrk X$ admits an infimum in $L'$
and:
:$\map \inf {f \sqbrk X} = \map f {\inf X}$
{{qed|lemma}}
Theorem: Moser's Circle Problem

Let $n$ points be marked on the circumference of a circle $C$.
Let chords be drawn between each pair of these points.
For each $n$, the maximum number $\map C n$ of regions into which $C$ can be divided is as follows:
:{| border="1"
|-
! align="right" style = "padding: 2px 10px" | $n$ 
! align="right" style = "padding: 2px 10px" | $\map C n$ 
|-
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2px 10px" | $1$
|-
| align="right" style = "padding: 2px 10px" | $2$
| align="right" style = "padding: 2px 10px" | $2$
|-
| align="right" style = "padding: 2px 10px" | $3$
| align="right" style = "padding: 2px 10px" | $4$
|-
| align="right" style = "padding: 2px 10px" | $4$
| align="right" style = "padding: 2px 10px" | $8$
|-
| align="right" style = "padding: 2px 10px" | $5$
| align="right" style = "padding: 2px 10px" | $16$
|-
| align="right" style = "padding: 2px 10px" | $6$
| align="right" style = "padding: 2px 10px" | $31$
|-
| align="right" style = "padding: 2px 10px" | $7$
| align="right" style = "padding: 2px 10px" | $57$
|-
| align="right" style = "padding: 2px 10px" | $8$
| align="right" style = "padding: 2px 10px" | $99$
|-
| align="right" style = "padding: 2px 10px" | $9$
| align="right" style = "padding: 2px 10px" | $163$
|-
| align="right" style = "padding: 2px 10px" | $10$
| align="right" style = "padding: 2px 10px" | $256$
|}
and in general:
:$\map C n = 1 + \dbinom n 2 + \dbinom n 4$

Proof:
:300px $\quad$ 300px $\quad$ 300px
:300px $\quad$ 300px $\quad$ 300px
{{proof wanted}}
{{Namedfor|Leo Moser|cat = Moser}}
Theorem: Prime Factors of 2^64 - 1

The prime decomposition of $2^{64} - 1$ is given by:
:$2^{64} - 1 = 3 \times 5 \times 17 \times 257 \times 641 \times 65 \, 537 \times 6 \, 700 \, 417$

Proof:
{{begin-eqn}}
{{eqn | l = 2^{64} - 1
      | r = \paren {2^{32} }^2 - 1
      | c = Power of Power
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{32} - 1}
      | c = Difference of Two Squares
}}
{{eqn | r = \paren {2^{32} + 1} \paren {\paren {2^{16} }^2 - 1}
      | c = Power of Power
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{16} + 1} \paren {2^{16} - 1}
      | c = Difference of Two Squares
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{16} + 1} \paren {\paren {2^8}^2 - 1}
      | c = Power of Power
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{16} + 1} \paren {2^8 + 1} \paren {2^8 - 1}
      | c = Difference of Two Squares
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{16} + 1} \paren {2^8 + 1} \paren {\paren {2^4}^2 - 1}
      | c = Power of Power
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{16} + 1} \paren {2^8 + 1} \paren {2^4 + 1} \paren {2^4 - 1}
      | c = Difference of Two Squares
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{16} + 1} \paren {2^8 + 1} \paren {2^4 + 1} \paren {\paren {2^2}^2 - 1}
      | c = Power of Power
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{16} + 1} \paren {2^8 + 1} \paren {2^4 + 1} \paren {2^2 + 1} \paren {2^2 - 1}
      | c = Difference of Two Squares
}}
{{eqn | r = \paren {2^{32} + 1} \paren {2^{16} + 1} \paren {2^8 + 1} \paren {2^4 + 1} \paren {2^2 + 1} \paren {2 + 1} \paren {2 - 1}
      | c = Difference of Two Squares
}}
{{eqn | r = \paren {2^{\paren {2^5} } + 1} \paren {2^{\paren {2^4} } + 1} \paren {2^{\paren {2^3} } + 1} \paren {2^{\paren {2^2} } + 1} \paren {2^{\paren {2^1} } + 1} \paren {2^{\paren {2^0} } + 1}
}}
{{end-eqn}}
From Sequence of Fermat Primes:
{{begin-eqn}}
{{eqn | l = 2^{\paren {2^0} } + 1
      | r = 3
}}
{{eqn | l = 2^{\paren {2^1} } + 1
      | r = 5
}}
{{eqn | l = 2^{\paren {2^2} } + 1
      | r = 17
}}
{{eqn | l = 2^{\paren {2^3} } + 1
      | r = 257
}}
{{eqn | l = 2^{\paren {2^4} } + 1
      | r = 65 \, 537
}}
{{end-eqn}}
all of which are Fermat primes.
Then:
{{begin-eqn}}
{{eqn | l = 2^{\paren {2^5} } + 1
      | r = 4 \, 294 \, 967 \, 297
      | c = Sequence of Fermat Numbers
}}
{{eqn | r = 641 \times 6 \, 700 \, 417
      | c = Prime Decomposition of $5$th Fermat Number
}}
{{end-eqn}}
Hence the result.
{{qed}}
Category:18,446,744,073,709,551,615
4ex0skqxwwau45hiss3a90x14q2j2za
Theorem: Prime Decomposition of 5th Fermat Number

The prime decomposition of the $5$th Fermat number is given by:
{{begin-eqn}}
{{eqn | l = 2^{\paren {2^5} } + 1
      | r = 4 \, 294 \, 967 \, 297
      | c = Sequence of Fermat Numbers
}}
{{eqn | r = 641 \times 6 \, 700 \, 417
      | c = 
}}
{{eqn | r = \paren {5 \times 2^7 + 1} \times \paren {3 \times 17449 \times 2^7 + 1}
      | c = 
}}
{{end-eqn}}

Proof:
From Divisor of Fermat Number, if $2^{\paren {2^n} } + 1$ has a divisor, it will be in the form:
:$k \, 2^{n + 2} + 1$
In the case of $n = 5$, a divisor of $2^{\paren {2^n} } + 1$ is then of the form:
:$k \, 2^7 + 1 = k \times 128 + 1$
Further, such a number will (for small $k$ at least) be prime, otherwise it will itself have a divisor which is not of that form.
Thus we investigate:
{{begin-eqn}}
{{eqn | l = 1 \times 128 + 1
      | r = 129
      | rr= = 3 \times 43
      | c = which is not prime
}}
{{eqn | l = 2 \times 128 + 1
      | r = 257
      | c = which is prime
}}
{{eqn | l = 3 \times 128 + 1
      | r = 385
      | rr= = 5 \times 7 \times 11
      | c = which is not prime
}}
{{eqn | l = 4 \times 128 + 1
      | r = 513
      | rr= = 3^3 \times 19
      | c = which is not prime
}}
{{eqn | l = 5 \times 128 + 1
      | r = 641
      | c = which is prime
}}
{{eqn | l = 6 \times 128 + 1
      | r = 769
      | c = which is prime
}}
{{end-eqn}}
We have that:
{{begin-eqn}}
{{eqn | l = 2^{32} - 1
      | r = \paren {2^{16} + 1} \paren {2^{16} - 1}
      | c = Difference of Two Squares
}}
{{eqn | r = \paren {2^{16} + 1} \paren {2^8 - 1} \paren {2^8 + 1}
      | c = 
}}
{{end-eqn}}
But
:$2^8 + 1 = 257$
and so:
:$2^{\paren {2^5} } - 1 \equiv 0 \pmod {257}$
which means:
:$2^{\paren {2^5} } + 1 \equiv 2 \pmod {257}$
so there is no need to do a trial division of $2^{\paren {2^5} } + 1$ by $257$.
Now dividing $4 \, 294 \, 967 \, 297$ by $641$ we find:
<pre>
          6 700 417
    ---------------
641 ) 4 294 967 297
      3 846
      -----
        448 9
        448 7
        -----
            267 2
            256 4
            -----
             10 89
              6 41
             -----
              4 487
              4 487
              =====
</pre>
and the divisor has been found.
{{qed}}
Theorem: Prime Decomposition of 5th Fermat Number

The prime decomposition of the $5$th Fermat number is given by:
{{begin-eqn}}
{{eqn | l = 2^{\paren {2^5} } + 1
      | r = 4 \, 294 \, 967 \, 297
      | c = Sequence of Fermat Numbers
}}
{{eqn | r = 641 \times 6 \, 700 \, 417
      | c = 
}}
{{eqn | r = \paren {5 \times 2^7 + 1} \times \paren {3 \times 17449 \times 2^7 + 1}
      | c = 
}}
{{end-eqn}}

Proof:
Note the remarkable coincidence that $2^4 + 5^4 = 2^7 \cdot 5 + 1 = 641$.
First we eliminate $y$ from $x^4 + y^4 = x^7 y + 1 = 0$:
{{begin-eqn}}
{{eqn | l = x^4 + y^4
      | r = x^7 y + 1 = 0
      | c = 
}}
{{eqn | ll= \leadsto
      | l = -x^4
      | r = y^4
      | c = 
}}
{{eqn | ll= \leadsto
      | l = x^{28} \paren {-x^4} + 1
      | r = 0
      | c = 
}}
{{eqn | ll= \leadsto
      | l = x^{32}
      | r = -1
      | c = 
}}
{{end-eqn}}
Now we use the above result for $x = 2$ and $y = 4$ in modulo $641$:
{{begin-eqn}}
{{eqn | l = 2^4 + 5^4
      | o = \equiv
      | r = 0
      | rr= \pmod {641}
      | c = 
}}
{{eqn | l = 2^7 \cdot 5
      | o = \equiv
      | r = -1
      | rr= \pmod {641}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 2^{28} \paren {-2^4}
      | o = \equiv
      | r = -1
      | rr= \pmod {641}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 2^{32}
      | o = \equiv
      | r = -1
      | rr= \pmod {641}
      | c = 
}}
{{eqn | ll= \leadsto
      | l = 2^{32} + 1
      | o = \equiv
      | r = 0
      | rr= \pmod {641}
      | c = 
}}
{{end-eqn}}
Thus $2^{\paren {2^5} } + 1 = 6 \, 700 \, 417 \times 641$ and hence is not prime.
{{qed}}
Theorem: Power of 2 is Difference between Two Powers

Let $n \in \Z_{>0}$ be a power of $2$.
Then $n$ is the difference between powers of two positive integers greater than or equal to $2$.
{{questionable|This is so trivial I wonder whether something got lost in translation.}}

Proof:
$2^k = 2^{k+1} - 2^k$
{{qed}}
Theorem: Smallest Sequence of Three Consecutive Semiprimes

The smallest triple of consecutive semiprimes is:
:$33, 34, 35$

Proof:
We have:
{{begin-eqn}}
{{eqn | l = 33
      | r = 3 \times 11
}}
{{eqn | l = 34
      | r = 2 \times 17
}}
{{eqn | l = 35
      | r = 5 \times 7
}}
{{end-eqn}}
It can be seen from the sequence of semiprimes that there exist no smaller such triples.
{{qed}}
Theorem: Integers not Sum of Distinct Triangular Numbers

The sequence of integers which cannot be expressed as the sum of distinct triangular numbers is:
:$2, 5, 8, 12, 23, 33$
{{OEIS|A053614}}

Proof:
It will be proved that the largest integer which cannot be expressed as the sum of distinct triangular numbers is $33$.
The remaining integers in the sequence can be identified by inspection.
We prove this using a variant of Second Principle of Mathematical Induction.
Let $\map P n$ be the proposition:
:$n$ can be expressed as the sum of distinct triangular numbers.
Theorem: Powers of 2 and 5 without Zeroes

The following $n \in \Z$ are such that both $2^n$ and $5^n$ have no zeroes in their decimal representation:
:$0, 1, 2, 3, 4, 5, 6, 7, 9, 18, 33$
{{OEIS|A007496}}

Proof:
:{| border="1"
|-
! align="center" style = "padding: 2px 10px" | $n$ 
! align="center" style = "padding: 2px 10px" | $2^n$ 
! align="center" style = "padding: 2px 10px" | $5^n$ 
|-
| align="right" style = "padding: 2px 10px" | $0$
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2px 10px" | $1$
|-
| align="right" style = "padding: 2px 10px" | $1$
| align="right" style = "padding: 2px 10px" | $2$
| align="right" style = "padding: 2px 10px" | $5$
|-
| align="right" style = "padding: 2px 10px" | $2$
| align="right" style = "padding: 2px 10px" | $4$
| align="right" style = "padding: 2px 10px" | $25$
|-
| align="right" style = "padding: 2px 10px" | $3$
| align="right" style = "padding: 2px 10px" | $8$
| align="right" style = "padding: 2px 10px" | $125$
|-
| align="right" style = "padding: 2px 10px" | $4$
| align="right" style = "padding: 2px 10px" | $16$
| align="right" style = "padding: 2px 10px" | $625$
|-
| align="right" style = "padding: 2px 10px" | $5$
| align="right" style = "padding: 2px 10px" | $32$
| align="right" style = "padding: 2px 10px" | $3125$
|-
| align="right" style = "padding: 2px 10px" | $6$
| align="right" style = "padding: 2px 10px" | $64$
| align="right" style = "padding: 2px 10px" | $15 \, 625$
|-
| align="right" style = "padding: 2px 10px" | $7$
| align="right" style = "padding: 2px 10px" | $128$
| align="right" style = "padding: 2px 10px" | $78 \, 125$
|-
| align="right" style = "padding: 2px 10px" | $9$
| align="right" style = "padding: 2px 10px" | $512$
| align="right" style = "padding: 2px 10px" | $1 \, 953 \, 125$
|-
| align="right" style = "padding: 2px 10px" | $18$
| align="right" style = "padding: 2px 10px" | $262 \, 144$
| align="right" style = "padding: 2px 10px" | $3 \, 814 \, 697 \, 265 \, 625$
|-
| align="right" style = "padding: 2px 10px" | $33$
| align="right" style = "padding: 2px 10px" | $8 \, 589 \, 934 \, 592$
| align="right" style = "padding: 2px 10px" | $116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125$
|}
It is probable that $n = 33$ is the final instance.
{{qed}}
Theorem: Vector Cross Product Distributes over Addition

The vector cross product is distributive over addition.
That is, in general:
:$\mathbf a \times \paren {\mathbf b + \mathbf c} = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}$
for $\mathbf a, \mathbf b, \mathbf c \in \R^3$.

Proof:
Let:
:$\mathbf a = \begin {bmatrix} a_x \\ a_y \\a_z \end {bmatrix}$, $\mathbf b = \begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix}$, $\mathbf c = \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix}$
be vectors in $\R^3$.
Then:
{{begin-eqn}}
{{eqn | l = \mathbf a \times \paren {\mathbf b + \mathbf c}
      | r = \begin {bmatrix} a_x \\ a_y \\a_z \end {bmatrix} \times \paren {\begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix} + \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix} }
}}
{{eqn | r = \begin {bmatrix} a_x \\ a_y \\a_z \end {bmatrix} \times {\begin {bmatrix} b_x + c_x \\ b_y + c_y \\ b_z + c_z \end {bmatrix} }
      | c = {{Defof|Vector Sum}}
}}
{{eqn | r = \begin {bmatrix} a_y \paren {b_z + c_z} - a_z \paren {b_y + c_y} \\ a_z \paren {b_x + c_x} - a_x \paren {b_z + c_z} \\ a_x \paren {b_y + c_y} - a_y \paren {b_x + c_x} \end {bmatrix}
      | c = {{Defof|Vector Cross Product}}
}}
{{eqn | r = \begin {bmatrix} a_y b_z + a_y c_z - a_z b_y - a_z c_y \\ a_z b_x + a_z c_x - a_x b_z - a_x c_z \\ a_x b_y + a_x c_y - a_y b_x - a_y c_x \end {bmatrix}
      | c = Real Multiplication Distributes over Addition
}}
{{eqn | r = \begin {bmatrix} a_y b_z - a_z b_y + a_y c_z - a_z c_y \\ a_z b_x - a_x b_z + a_z c_x - a_x c_z \\ a_x b_y - a_y b_x + a_x c_y - a_y c_x \end {bmatrix}
      | c = Real Addition is Commutative
}}
{{eqn | r = \begin {bmatrix} a_y b_z - a_z b_y \\ a_z b_x - a_x b_z \\ a_x b_y - a_y b_x \end {bmatrix} + \begin {bmatrix} a_y c_z - a_z c_y \\ a_z c_x - a_x c_z \\ a_x c_y - a_y c_x \end {bmatrix}
      | c = {{Defof|Vector Sum}}
}}
{{eqn | r = \paren {\begin {bmatrix}a_x \\ a_y \\ a_z \end {bmatrix} \times \begin {bmatrix} b_x \\ b_y \\ b_z \end {bmatrix} } + \paren {\begin {bmatrix} a_x \\ a_y \\ a_z \end {bmatrix} \times \begin {bmatrix} c_x \\ c_y \\ c_z \end {bmatrix} }
      | c = {{Defof|Vector Cross Product}}
}}
{{eqn|r = \paren {\mathbf a \times \mathbf b} + \paren {\mathbf a \times \mathbf c}
}}
{{end-eqn}}
{{qed}}