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Vincent Granville’s proposed representations are conjectures, not established theorems. The best-known claim says that every non-square integer z can be written as z = x2 + y, with x an integer and y prime. Granville’s article presents a heuristic counting argument for why representations should occur frequently, but that argument does not prove the claim for all integers. The article also reports a separate floor-power conjecture. The article does not establish that either conjecture has been proved or disproved.
What Granville’s first conjecture says
The statement concerns every integer that is not a perfect square. For each such z, it proposes finding an integer x and a prime number y satisfying:
z = x2 + y.
Because x may be negative, the square term is effectively determined by the nonnegative value |x|. The prime summand must be positive under the usual definition of a prime. Perfect squares are excluded because the claim is aimed at numbers whose difference from a suitable square is prime.
How to test one integer
For a particular non-square z, a finite check is straightforward: examine integer values of x with x2 < z, calculate y = z − x2, and test whether y is prime. Finding one valid pair verifies that particular z; failing to find one in the tested range does not prove that no pair exists unless every possible x has been covered.
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Why this is not a theorem
The article explicitly describes its reasoning as heuristic. Its central idea is an area-counting estimate: solutions are considered below a boundary described using a curve of the form z = x2 + w log w, and the resulting count is expected to grow on average. Such estimates can suggest that representations should be plentiful, but an average-growth prediction is not a proof that every individual non-square has a representation.
- A heuristic estimates likely behavior; it does not establish a universal quantifier such as “for every non-square integer.”
- Prime values are irregular, so an expected number of solutions can coexist with isolated exceptions.
- A proof would need to rule out every possible counterexample, not merely show that representations become common on average.
Finite computation versus proof
The article asks whether the conjecture can be verified up to a very large value of z. A computation can provide evidence over a stated finite interval, but it cannot settle the assertion for all integers.
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What a computation can establish
- That each tested non-square in a specified range has at least one displayed decomposition.
- Which values fail within that exact range and under the exact primality test used.
- Examples of multiple decompositions, if the program records all candidates rather than stopping at the first.
What it cannot establish
- That no exception exists beyond the tested upper limit.
- That an exception list reported by a question author is exhaustive without a reproducible, complete search description.
- That the heuristic argument has become a proof.
The article reports an exception set and a computational range attributed to the author of the question. Those observations should be treated as reported finite checks, not independently verified results or proof of exhaustiveness.
The separate floor-power conjecture
The article gives another conjectural representation:
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Every integer is representable as ⌊xc⌋ + ⌊yc⌋
Here x and y are positive integers, and c is a positive constant subject to the bound written in the article as c < log₂₂(63). The article does not establish the meaning of that bound beyond its displayed notation, nor does it prove that a qualifying constant exists or that the representation works for every integer.
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- Used Book in Good Condition
This proposal differs from the square-plus-prime claim in both its inputs and its summands:
| Proposal | Numbers being represented | Summands | Status in the available material |
|---|---|---|---|
| Square plus prime | Every non-square integer z | x2 and a prime y | Conjecture supported by heuristic discussion; not proved in the excerpt |
| Floor powers | All integers | ⌊xc⌋ and ⌊yc⌋ | Conjecture; the excerpt does not establish it |
How this relates to Waring’s problem
Waring’s problem asks whether, for each fixed exponent k, every positive integer can be represented as a sum of a bounded number of kth powers. The associated existence results are part of established number theory. They concern fixed-power summands and a bounded number of terms.
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Granville’s proposals use different ingredients. The first combines one square with a prime and excludes perfect squares from the target domain. The second uses two floor powers with a constant exponent parameter. The article does not provide a formal theorem showing that either statement is a generalization of Waring’s problem in the technical sense. The title’s “generalization” should therefore be read as a motivating comparison, not as an established classification.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.What is known about the article’s date and status
DataScienceCentral’s archive dates “Number Theory: Nice Generalization of the Waring Conjecture” to October 1, 2018. That date identifies the archived article; it does not make the proposals current results. The article does not confirm a later proof, disproof, or definitive contemporary status for either conjecture.
How to read claims about verification
When you encounter a statement that the square-plus-prime claim was “verified” to a large limit, ask three questions:
- What was the exact upper bound? A finite limit leaves all larger integers untreated.
- Were all admissible values of x searched? For a given z, the search must cover every relevant square below z.
- Is the primality test and implementation documented? Without reproducible details, the result is evidence rather than an independently checkable theorem.
A proof would require a mathematical argument covering the infinite domain. Neither the reported exception search nor the area-counting heuristic supplies that missing step.
Bottom line for readers
The square-plus-prime representation and the floor-power representation are interesting number-theory conjectures associated with Granville’s 2018 article. The first says non-squares should equal a square plus a prime; the second says integers should equal two floor powers for a suitable constant. The article presents motivation and finite computational observations, not proof. Treat any claimed test range as limited evidence, and do not confuse these proposals with the established results surrounding Waring’s problem.
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