Squarefree quartic values and the large-prime obstacle

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Family 020 · Number theory
Importance 4/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

An integer is squarefree if no prime square divides it. The numbers $6$ and $10$ are squarefree; $12$ is not, because $4\mid12$. Among all positive integers, squarefree numbers have a familiar positive density. But what happens in a sparse polynomial sequence such as

$$1^4+2,\ 2^4+2,\ 3^4+2,\ldots?$$

There is no obvious reason a positive fraction of these values should avoid every prime square at once. Testing divisibility by a fixed finite list of primes is straightforward. The difficulty is excluding square factors from very large primes, whose squares can divide a polynomial value even when the input is much smaller than the prime.

Family 020 claims the expected squarefree density for every irreducible quartic with no local obstruction. More generally, it fills the low-degree gaps for values free of $(d-2)$-th prime powers. Combined with an established higher-degree theorem, this yields that density for every degree $d\ge4$.

2. Background and essential definitions

For an integer $k\ge2$, a nonzero integer is $k$-power-free if no prime power $p^k$ divides it. Negative values are treated through divisibility in exactly the same way. Zero is excluded because every prime power divides zero. The case $k=2$ is squarefreeness.

For a polynomial $f\in\mathbb Z[x]$, define

$$\rho_f(q)=\#\{a\bmod q:f(a)\equiv0\pmod q\}.$$

This counts the residue classes that fail the divisibility test modulo $q$. A necessary condition for $k$-power-free values is

$$\rho_f(p^k)<p^k\qquad\text{for every prime }p.$$

If equality holds for one prime, every input gives a value divisible by that same $p^k$. Such a prime power is a fixed divisor obstruction.

When no obstruction occurs, the predicted density is

$$c_{f,k}=\prod_p\left(1-\frac{\rho_f(p^k)}{p^k}\right).$$

For any finite set of primes, the Chinese remainder theorem makes this product exact: residue conditions at different prime powers combine independently. The hard step is justifying the passage to infinitely many primes while the input bound grows.

An irreducible polynomial over $\mathbb Q$ cannot be factored there into two positive-degree polynomials. In characteristic zero it has distinct complex roots. This separability ensures that, outside finitely many exceptional primes, roots modulo $p$ lift uniquely to roots modulo $p^k$, and $\rho_f(p^k)\le\deg f$. Thus the infinite product converges to a positive number once every local factor is positive.

Write

$$S_{f,k}(X)=\#\{1\le n\le X:f(n)\text{ is }k\text{-power-free}\}.$$

An asymptotic $S_{f,k}(X)=c_{f,k}X+o(X)$ says that the proportion approaches $c_{f,k}$. The notation $o(X)$ means an error whose ratio to $X$ tends to zero. It does not, by itself, promise a particular rate.

3. Development history and previous partial results

Classical work of Ricci obtained power-free asymptotics when the excluded power was at least the polynomial degree. Erdős proved infinitude in the $k=d-1$ range, and Hooley obtained its asymptotic form. The cubic squarefree case belongs to that range. Erdős explicitly highlighted $n^4+2$ as an unresolved quartic example in 1953 and returned to the degree-minus-two obstacle in his later survey, both cited by the manuscript.

Nair and Huxley–Nair extended the accessible range. Heath-Brown’s determinant method reached $k\ge(3d+2)/4$. Browning improved this to

$$k\ge\frac{3d+1}{4},$$

which already includes $k=d-2$ for $d\ge9$. See Browning’s 2011 paper. This is why the new manuscript concentrates on degrees four through eight rather than claiming that every degree requires a new argument.

Special polynomial shapes can admit stronger results. Heath-Brown obtained power-saving asymptotics for irreducible binomials $x^d+c$ in a better range, and Reuss obtained power-saving results at exponent $d-1$ for general fixed polynomials. The determinant and number-field techniques developed in these works are direct methodological predecessors.

There are also results with different quantifiers. Squarefree binary forms vary two integer inputs, while the present problem fixes one input to one. Averaging over many coefficient choices is different from proving a theorem for each fixed polynomial. Neither automatically resolves an individual quartic sequence.

Granville showed that the abc conjecture would give the predicted squarefree-value density for general separable polynomials satisfying the local condition. His primary paper explains that conditional landscape. The present manuscript states an unconditional result in its degree and power range, without using abc.

4. What the new manuscripts claim

The new low-degree theorem

Theorem 1.1 concerns an integer polynomial irreducible over $\mathbb Q$, of degree $4\le d\le8$, with $k=d-2$. Assuming no fixed prime $k$-th power divisor, it claims

$$S_{f,k}(X)=c_{f,k}X+o_f(X),\qquad c_{f,k}>0.$$

The powers are as follows:

Degree $d$	Excluded prime power
4	$p^2$: squarefree values
5	$p^3$: cube-free values
6	$p^4$
7	$p^5$
8	$p^6$

Inputs are positive integers. The polynomial is fixed, with no bound imposed on its coefficient sizes. It need not be primitive, meaning its coefficients may have a common factor, and its leading coefficient may be negative. The exact local factors retain this content and sign generality.

Combining this theorem with Browning’s higher-degree result gives Corollary 1.2: the same $(d-2)$-power-free density for every irreducible degree $d\ge4$ satisfying the corresponding local condition.

Products and simultaneous values

For fixed $k\ge2$, Corollary 1.3 treats a separable polynomial whose nonconstant irreducible factors each have degree at most $k+2$. Here separable means no repeated nonconstant factor. Under the joint local admissibility condition, its whole value is $k$-power-free with the predicted density.

The corollary also treats finitely many fixed polynomials $F_1,\ldots,F_r$ of this kind, requiring each value separately to be $k$-power-free. Let

$$\Omega_p=\{a\bmod p^k:p^k\mid F_i(a)\text{ for at least one }i\}.$$

If $|\Omega_p|<p^k$ for all primes, the density is

$$\prod_p\left(1-\frac{|\Omega_p|}{p^k}\right)>0.$$

Different polynomials may share factors. Requiring each value separately to be power-free is different from requiring their product to be power-free. A shared prime can appear once in each of two squarefree values and twice in their product. The local union $\Omega_p$ correctly captures the separate-value question.

Two concrete quartics

The theorem includes $x^4+2$, irreducible by Eisenstein’s criterion at two, and the cyclotomic polynomial $x^4+1$. Both satisfy the local obstruction condition.

For $x^4+1$, the manuscript works out the exact density:

$$c_8=\prod_{p\equiv1\pmod8}\left(1-\frac4{p^2}\right)>\frac34.$$

There are no roots modulo four. For odd $p$, a root modulo $p$ has multiplicative order eight, so roots exist precisely when $p\equiv1\pmod8$, with four simple roots then lifting modulo $p^2$. Corollary 12.1 consequently claims that, for all sufficiently large real $N$, at least $N/2$ integers in $[N,2N]$ give squarefree values of $n^4+1$.

5. Why the advance matters

Provisional importance: 4/5. Taken as stated, the quartic theorem resolves a classical barrier in polynomial squarefreeness, including Erdős’s example $n^4+2$. The family also completes the degree-minus-two range by connecting the new low-degree argument to established higher-degree work.

The result gives the full predicted positive density, stronger than merely proving infinitely many examples. Its hypotheses match the unavoidable local obstruction for the stated class of polynomials.

Methodologically, the manuscript combines number-field factorization with determinant estimates that adapt their auxiliary primes as the geometry changes. Its special quartic analysis explains why the first unresolved degree needs more than a routine substitution into general estimates.

6. The main idea

Separate small primes from large primes

Fix a cutoff $Y$. Divisibility by primes $p\le Y$ is handled exactly by residue classes and the Chinese remainder theorem. Medium primes can be bounded using root counts and lifting. The hard tail is

$$\#\{X<n\le2X:p^k\mid f(n)\text{ for some prime }p>X\}.$$

For primitive irreducible polynomials with positive leading coefficient in degrees four through eight, Proposition 1.4 claims that this is $O_f(X^{1-\delta})$ for some $\delta>0$. It does not require the local admissibility condition.

This power saving is an intermediate large-prime estimate. The final density proof first lets $X$ grow with $Y$ fixed, then lets $Y$ grow. The manuscript does not promote that procedure to a power-saving error for the complete asymptotic.

Factor the equation inside a number field

If $p^k\mid f(n)$, write $f(n)=yp^k$. The triples lie on the surface

$$f(x)=yz^k.$$

For some prime-size ranges, a weighted determinant estimate on this surface suffices. In the harder range, choose a root $\theta$ of $f$ and work in $K=\mathbb Q(\theta)$. Ideal factorization and units lead to relations of the form

$$\alpha^k\beta=\mu(n-\theta),$$

with integral field elements $\alpha,\beta$ and a fixed denominator-clearing integer $\mu$. Units balance their sizes at the different embeddings. Coordinates in an integral basis then place $\alpha$ and $\beta$ in paired boxes.

Their directions are not independent: near a balanced point, either direction determines the other analytically once $1/n$ is specified. This is the extra arithmetic structure missing from the original surface equation.

Force the points onto smaller algebraic sets

The determinant method arranges monomial values at integer points into a matrix. Integrality or divisibility gives a lower bound for a nonzero determinant; sufficiently strong analytic bounds force it to vanish. Vanishing produces an auxiliary polynomial containing the points.

The manuscript repeatedly cuts the dimension in paired boxes. It distinguishes equations involving both coordinate groups from those involving only one group, because the available Taylor monomials differ. Reduced lattice bases shrink the coordinates, while unusually short vectors restrict how many boxes can contain them.

The auxiliary varieties depend on earlier cuts, so a fixed list of good primes cannot simply be assumed. The adaptive-prime counting argument controls primes where smooth coordinates fail as the construction proceeds.

Quartics require sharper surface and curve geometry. Exceptional projections and low-degree maps are treated separately. One limiting case involves degree-five polynomials with prescribed critical values; their relevant integer values form a single small exceptional set removed before summing over boxes. This avoids repeatedly charging the same exceptional inputs.

7. Scope, limitations, and open questions

The general degree-$d$ conclusion excludes $p^{d-2}$. It does not prove squarefree values for every degree. For example, the degree-five theorem is cube-free, not squarefree. The unrestricted squarefree-value conjecture for higher-degree irreducible polynomials remains beyond this statement.

The polynomials, their number in a simultaneous family, and the exponent $k$ are fixed. No uniform coefficient bound or uniform error term is asserted. The final asymptotic has an $o(X)$ error, even though a particular large-prime tail has a power saving.

The local condition is indispensable. It must be checked for the whole polynomial or the simultaneous family, rather than inferred from separate factors without accounting for shared residue obstructions. Repeated factors are excluded from the separable-product corollary.

Further questions include effective rates in the full asymptotic, squarefree values in higher degrees, uniformity over coefficient families and values at prime rather than unrestricted integer inputs.

8. Sources and further reading

Theorem 1.1 states the new degree-four-through-eight result; Corollaries 1.2–1.3 give the higher-degree and simultaneous consequences. Proposition 1.4 isolates the large-prime estimate. Section 12 supplies the worked $n^4+1$ example and its Euler product.

Browning’s and Granville’s papers linked in section 3 explain the established higher-degree range and the abc-based general prediction. The manuscript’s bibliography identifies the earlier Erdős, Hooley, Nair, Heath-Brown and Reuss contributions. Optional background on squarefree integers and the Chinese remainder theorem helps interpret the local factors.

Source manuscripts

• M1. OpenAI, Squarefree values of quartics and power-free values of polynomials (2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.