The empty sequence

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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

• Conjectured empty as a result of Goldbach conjecture:

Even integers $\scriptstyle n\,\geq \,6\,$ which are not the sum of at most 2 odd primes ("strong" Goldbach conjecture).

Odd integers $\scriptstyle n\,\geq \,9\,$ 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 (1770):

Positive integers which are not the sum of at most $\scriptstyle g(n)\,$ $\scriptstyle n\,$^th powers of positive integers, where:

$g(n)=(2^{n}-2)+{\bigg \lfloor }{{{\bigg (}{3 \over 2}{\bigg )}}^{n}}{\bigg \rfloor }=2\,(2^{n-1}-1)+{\bigg \lfloor }{{{\bigg (}{3 \over 2}{\bigg )}}^{n}}{\bigg \rfloor },\,$

with $\scriptstyle g(2)\,=\,4,\,g(3)\,=\,9,\,g(4)\,=\,19,\,g(5)\,=\,37,\,g(6)\,=\,73\,$ all proved, although $\scriptstyle g(7)\,=\,143\,$ is not yet proved.

• Conjectured empty as a result of Pollock's conjecture (1850) about the order of basis of the (3-dimensional) Platonic numbers:

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:

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:

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:

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:

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:

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):

Positive integers $\scriptstyle n\,$ such that $\scriptstyle n^{k}\,=\,a^{k}+b^{k},\,k\,\geq \,3,\,a\,>\,0,\,b\,>\,0.\,$

• Proved empty as a result of Cauchy's proof (1813) of Fermat's polygonal number theorem (proposed in 1638, proof never found):

Positive integers $\scriptstyle n\,$ that are not the sum of at most $\scriptstyle k\,$ (not necessarily distinct) $\scriptstyle k\,$-gonal numbers.

• Proved empty as a result of Mihăilescu's proof (published 2004) of Catalan's Conjecture (1844):

Positive integers $\scriptstyle x\,$ and $\scriptstyle y\,$ such that $\scriptstyle x^{p}-y^{q}\,=\,\pm 1,\,x\,\geq \,3,\,y\,\geq \,3,\,p\,\geq \,2,\,q\,\geq \,2.\,$

See also

• 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!)