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# The empty sequence

The empty sequence (as there is only one empty sequence) is the "solution" (so to speak) of a countably infinite number of unsatisfiable sequence definitions (or enunciable problems without solutions). Any sequence whose definition is not satisfiable results in the empty sequence. Some definitions are conjectured not satisfiable (conjectured empty) while others have been proved not satisfiable (proved empty).

It seems fitting to assign sequence A000000 to the empty sequence since it is the only sequence with cardinality 0. But the A-number A000000 is inadmissible in the Main OEIS: since that sequence has no terms, lookup programs would not be able to handle it.

## Conjectured empty

Even integers ${\displaystyle \scriptstyle n\,\geq \,6\,}$ which are not the sum of at most 2 odd primes ("strong" Goldbach conjecture).
Odd integers ${\displaystyle \scriptstyle n\,\geq \,9\,}$ which are not the sum of at most 3 odd primes ("weak" Goldbach conjecture).
• Conjectured empty as a result of the conjectured explicit formula by Dickson, Pillai, and Niven (1936) (cf. A002804) to Waring's problem[2] (1770):
Positive integers which are not the sum of at most ${\displaystyle \scriptstyle g(n)\,}$ ${\displaystyle \scriptstyle n\,}$th powers of positive integers, where:
${\displaystyle g(n)=(2^{n}-2)+{\bigg \lfloor }{{{\bigg (}{3 \over 2}{\bigg )}}^{n}}{\bigg \rfloor }=2\,(2^{n-1}-1)+{\bigg \lfloor }{{{\bigg (}{3 \over 2}{\bigg )}}^{n}}{\bigg \rfloor },\,}$
with ${\displaystyle \scriptstyle g(2)\,=\,4,\,g(3)\,=\,9,\,g(4)\,=\,19,\,g(5)\,=\,37,\,g(6)\,=\,73\,}$ all proved, although ${\displaystyle \scriptstyle g(7)\,=\,143\,}$ is not yet proved.
Positive integers which are not the sum of at most: 5 tetrahedral numbers, 7 octahedral numbers, 9 (proved) cubic numbers, 13 icosahedral numbers, 21 dodecahedral numbers;
while according to Hyun Kwang Kim's computer search (2002) the numerical evidence leads to:[5]
Positive integers which are not the sum of at most: 5 tetrahedral numbers, 7 octahedral numbers, 9 (proved) cubic numbers, 15 icosahedral numbers, 22 dodecahedral numbers.
• Conjectured empty as a result of Hyun Kwang Kim's computer search (2002) numerical evidence about the order of basis of the (4-dimensional) regular polychoron numbers:[6][5]
Positive integers which are not the sum of at most: 8 pentachoron numbers, 11 tetracross numbers, 19 (proved) tesseract numbers, 28 24-cell numbers,
125 600-cell numbers, 606 120-cell numbers.
• Conjectured empty as a result of Hyun Kwang Kim's computer search (2002) numerical evidence about the order of basis of the (5-dimensional) regular polyteron numbers:[7]
Positive integers which are not the sum of at most: 10 hexateron numbers, 14 pentacross numbers, 37 (proved) penteract numbers.
• Conjectured empty as a result of Hyun Kwang Kim's computer search (2002) numerical evidence about the order of basis of the (6-dimensional) regular polypeton numbers:[8]
Positive integers which are not the sum of at most: 13 heptapeton numbers, 19 hexacross numbers, 73 (proved) hexeract numbers.
• Conjectured empty as a result of Hyun Kwang Kim's computer search (2002) numerical evidence about the order of basis of the (7-dimensional) regular polyhexon numbers:[9]
Positive integers which are not the sum of at most: 15 octahexon numbers, 21 heptacross numbers, 143 hepteract numbers.

## Proved empty

• Proved empty as a result of Andrew Wiles' proof (final corrected proof published in 1995) of Fermat's last theorem (proposed in 1637, proof never found):[10]
Positive integers ${\displaystyle \scriptstyle n\,}$ such that ${\displaystyle \scriptstyle n^{k}\,=\,a^{k}+b^{k},\,k\,\geq \,3,\,a\,>\,0,\,b\,>\,0.\,}$
Positive integers ${\displaystyle \scriptstyle n\,}$ that are not the sum of at most ${\displaystyle \scriptstyle k\,}$ (not necessarily distinct) ${\displaystyle \scriptstyle k\,}$-gonal numbers.
• Proved empty as a result of Mihăilescu's proof (published 2004) of Catalan's Conjecture (1844):[12]
Positive integers ${\displaystyle \scriptstyle x\,}$ and ${\displaystyle \scriptstyle y\,}$ such that ${\displaystyle \scriptstyle x^{p}-y^{q}\,=\,\pm 1,\,x\,\geq \,3,\,y\,\geq \,3,\,p\,\geq \,2,\,q\,\geq \,2.\,}$

• Cf. A000000 (OEIS Wiki page; A000000 is NOT in the Main OEIS since it is NOT an admissible A-number, and the empty sequence is NOT searchable!)

## Notes

1. Eric W. Weisstein, Goldbach Conjecture, from MathWorld — A Wolfram Web Resource.
2. Eric W. Weisstein, Waring's Problem, from MathWorld — A Wolfram Web Resource.
3. The number of vertices of the five Platonic solids are 4 for tetrahedrons, 6 for octahedrons, 8 for hexahedrons (cubes), 12 for icosahedrons and 20 for dodecahedrons. Thus Pollock's conjecture amounts to the stipulation that the order of basis for each of the five (3-dimensional) Platonic solids is ${\displaystyle \scriptstyle V+1\,}$, where ${\displaystyle \scriptstyle V\,}$ is the number of vertices! One would hope that this nice pattern would hold, but Hyun Kwang Kim's computer search (2002) numerical evidence seems to lead away from it... Compare it with Fermat's polygonal number theorem which states that the order of basis of the (2-dimensional) ${\displaystyle \scriptstyle k\,}$-gonal numbers is ${\displaystyle \scriptstyle k\,}$, which is the number of vertices, whence the order of basis is ${\displaystyle \scriptstyle V+0\,}$. Hyun Kwang Kim's conjecture for the Platonic numbers thus states that the order of basis is ${\displaystyle \scriptstyle V+1\,}$ for tetrahedral numbers, octahedral numbers and hexahedral (cubic) numbers, ${\displaystyle \scriptstyle V+3\,}$ for icosahedral numbers and ${\displaystyle \scriptstyle V+2\,}$ for dodecahedral numbers. If you look at Hyun Kwang Kim's conjecture for the regular polychoron numbers, which gives the sequence ${\displaystyle \scriptstyle V+\{3,\,3,\,3,\,4,\,5,\,6\}\,}$ for the order of basis for the six polychoron numbers, an analogue for the order of basis for the five Platonic numbers might have been ${\displaystyle \scriptstyle V+\{1,\,1,\,1,\,2,\,3\}\,}$ instead of ${\displaystyle \scriptstyle V+\{1,\,1,\,1,\,3,\,2\}\,}$.
4. Pollock, Frederick, On the extension of the principle of Fermat's theorem on the polygonal numbers to the higher order of series whose ultimate differences are constant. With a new theorem proposed, applicable to all the orders, Abstracts of the Papers Communicated to the Royal Society of London, 5 (1850) pp. 922-924.
5. Hyun Kwang Kim, On Regular Polytope Numbers.
6. The number of vertices of the six regular polychorons are 5 for pentachorons (hypertetrahedrons), 8 for tetracross (hyperoctahedrons), 16 for tesseracts (hypercubes), 24 for 24-cell polychorons, 120 for 600-cell polychorons (hypericosahedrons) and 600 for 120-cell polychorons (hyperdodecahedrons). Thus Hyun Kwang Kim's conjecture amounts to saying that the order of basis for the six (4-dimensional) regular polychoron numbers is ${\displaystyle \scriptstyle V+3\,}$ for pentachoron numbers, tetracross numbers and tesseract numbers, ${\displaystyle \scriptstyle V+4\,}$ for 24-cell numbers, ${\displaystyle \scriptstyle V+5\,}$ for 600-cell polychorons and ${\displaystyle \scriptstyle V+6\,}$ for 120-cell polychorons, where ${\displaystyle \scriptstyle V\,}$ is the number of vertices. This gives the sequence ${\displaystyle \scriptstyle V+\{3,\,3,\,3,\,4,\,5,\,6\}\,}$ for the order of basis for the six regular polychoron numbers.
7. The number of vertices of the three regular polyterons are 6 for hexaterons (hypertetrahedrons), 10 for pentacross (hyperoctahedrons), 32 for penteracts (hypercubes). Thus Hyun Kwang Kim's conjecture amounts to saying that the order of basis for the three (5-dimensional) regular polyteron numbers is ${\displaystyle \scriptstyle V+4\,}$ for pentachoron numbers and pentacross numbers and ${\displaystyle \scriptstyle V+5\,}$ for penteract numbers. This gives the sequence ${\displaystyle \scriptstyle V+\{4,\,4,\,5\}\,}$ for the order of basis for the three regular polyteron numbers.
8. The number of vertices of the three regular polypetons are 7 for heptapetons (hypertetrahedrons), 12 for hexacross (hyperoctahedrons), 64 for hexeracts (hypercubes). Thus Hyun Kwang Kim's conjecture amounts to saying that the order of basis for the three (6-dimensional) regular polypeton numbers is ${\displaystyle \scriptstyle V+6\,}$ for heptapeton numbers, ${\displaystyle \scriptstyle V+7\,}$ for hexacross numbers and ${\displaystyle \scriptstyle V+9\,}$ for hexeract numbers. This gives the sequence ${\displaystyle \scriptstyle V+\{6,\,7,\,9\}\,}$ for the order of basis for the three regular polypeton numbers.
9. The number of vertices of the three regular polyhexons are 8 for octahexons (hypertetrahedrons), 14 for heptacross (hyperoctahedrons), 128 for hepteracts (hypercubes). Thus Hyun Kwang Kim's conjecture amounts to saying that the order of basis for the three (7-dimensional) regular polyhexon numbers is ${\displaystyle \scriptstyle V+7\,}$ for octahexon numbers and heptacross numbers and ${\displaystyle \scriptstyle V+15\,}$ for hepteract numbers. This gives the sequence ${\displaystyle \scriptstyle V+\{7,\,7,\,15\}\,}$ for the order of basis for the three regular polyhexon numbers.
10. Eric W. Weisstein, Fermat's Last Theorem, from MathWorld — A Wolfram Web Resource.
11. Eric W. Weisstein, Fermat's Polygonal Number Theorem, from MathWorld — A Wolfram Web Resource.
12. Eric W. Weisstein, Catalan's Conjecture, from MathWorld — A Wolfram Web Resource.