A Beatty sequence selects integers by taking the floor of a steadily increasing irrational multiple. When the selected integers are filtered for primality, the primes are not evenly spaced, yet their long-run count follows a precise density law. For an irrational slope α > 1, roughly a fraction 1 / α of all primes occur in the corresponding Beatty sequence.
Prime Density in an Irrational Beatty Sequence
πB(x) ~ π(x) / α ~ x / (α log x)
Here πB(x) counts primes not exceeding x that occur in the Beatty sequence, while π(x) is the ordinary prime counting function.
Beatty Sequences and What a Beatty Prime Means
For real numbers α > 1 and β, consider the non-homogeneous Beatty sequence
Bα,β = {⌊αn + β⌋ : n = 1, 2, 3, …}
The floor symbol ⌊x⌋ means the greatest integer not exceeding x. The parameter α controls how quickly the sequence advances, while β shifts the sequence.
When β = 0, the sequence has the simpler form
Bα = {⌊αn⌋ : n = 1, 2, 3, …}.
A Beatty prime is not a separate category of prime number. It is an ordinary prime that also happens to be a term of a specified Beatty sequence. A term can therefore satisfy the sequence rule and still be composite.
Two Different Questions
Sequence membership: Is p equal to ⌊αn + β⌋ for some positive integer n?
Primality: Does p have exactly two positive divisors, 1 and p?
A Golden-Ratio Beatty Sequence
The golden ratio provides a useful concrete example:
φ = (1 + √5) / 2 ≈ 1.6180339887
Bφ = {⌊φn⌋}
The first terms are:
1, 3, 4, 6, 8, 9, 11, 12, 14, 16, 17, 19, 21, 22, 24, 25, 27, 29, 30, 32, 33, 35, 37, 38, 40, 42, 43, 45, 46, 48, …
Among these early terms, the primes begin
3, 11, 17, 19, 29, 37, 43, 53, 59, 61, …
| n | φn | ⌊φn⌋ | Prime? |
|---|---|---|---|
| 1 | 1.618… | 1 | No |
| 2 | 3.236… | 3 | Yes |
| 3 | 4.854… | 4 | No |
| 4 | 6.472… | 6 | No |
| 5 | 8.090… | 8 | No |
| 6 | 9.708… | 9 | No |
| 7 | 11.326… | 11 | Yes |
| 11 | 17.798… | 17 | Yes |
| 12 | 19.416… | 19 | Yes |
| 18 | 29.124… | 29 | Yes |
The floor rule determines which integers enter the sequence. Whether one of those integers is prime is a separate divisibility question.
Test a Beatty Sequence Term
Enter an integer produced by a Beatty sequence to check whether it is prime, composite, or neither.
The Prime Count Follows the Density 1 / α
A Beatty sequence with α > 1 contains approximately one out of every α positive integers. More precisely, the number of its terms not exceeding x is asymptotic to
x / α.
For irrational α, the primes inside the sequence follow the same leading density relative to all primes:
πB(x) / π(x) → 1 / α
as x → ∞.
Because the prime number theorem gives π(x) ~ x / log x, this also gives
πB(x) ~ x / (α log x).
What 1 / α Actually Measures
The value 1 / α is the long-run fraction of all primes captured by the Beatty sequence. It is not the probability that an individual Beatty term is prime.
There are about x / α Beatty terms up to x and about x / (α log x) prime Beatty terms. Dividing the second estimate by the first shows that the fraction of Beatty terms that are prime is still on the scale of
1 / log x.
So an irrational Beatty sequence selects only a fixed fraction of the integers, but it does not create a growing advantage in primality.
Finite Counts Around the Golden-Ratio Prediction
For α = φ, the limiting share is
1 / φ ≈ 0.6180339887
so about 61.8% of all primes are expected to lie in Bφ in the asymptotic sense.
Exact counts at finite cutoffs fluctuate around that target.
| x | π(x) | π(x) / φ | Actual Beatty Primes | Observed Share |
|---|---|---|---|---|
| 100 | 25 | 15.45 | 14 | 56.00% |
| 1,000 | 168 | 103.83 | 100 | 59.52% |
| 10,000 | 1,229 | 759.56 | 769 | 62.57% |
| 100,000 | 9,592 | 5,928.18 | 5,973 | 62.27% |
The ratio is not required to equal 61.8034% at any particular cutoff. The theorem describes what happens as x grows without bound, so deviations at finite x are expected.
Why the Factor 1 / α Appears
An integer p belongs to Bα,β when some positive integer n satisfies
p ≤ αn + β < p + 1.
Equivalently, n must fall inside an interval of length 1 / α:
(p − β) / α ≤ n < (p + 1 − β) / α.
Because α > 1, that interval has length less than 1, so it contains at most one integer. Membership can therefore be described through the fractional positions generated by multiples of 1 / α.
When α is irrational, these fractional positions do not settle into a repeating finite cycle. Their distribution across primes is sufficiently uniform to produce the limiting fraction 1 / α. Proving that statement requires analytic number theory rather than treating primes as random numbers.
Integer density: Bα,β occupies asymptotically 1 / α of the positive integers.
Prime density: for irrational α, it also captures asymptotically 1 / α of the primes.
Why Irrational and Rational Slopes Must Be Separated
The irrational condition is mathematically meaningful. With a rational slope, the sequence can become locked into fixed residue classes, so the simple irrational density law cannot be applied blindly.
For example, with α = 2 and β = 0,
B2,0 = 2, 4, 6, 8, 10, …
Every term is even, so 2 is the only prime in the sequence.
Changing only the shift gives a completely different prime pattern:
B2,1 = 3, 5, 7, 9, 11, 13, …
This sequence contains every odd integer starting from 3 and therefore contains every odd prime.
Rational Slopes Need Congruence Analysis
A rational Beatty sequence can contain infinitely many primes, only finitely many primes, or be constrained by several residue classes. Its behavior depends on the slope and shift rather than on the factor 1 / α alone.
This is closely related to primes in arithmetic progressions and modular arithmetic. Once the slope is rational, congruence restrictions can decide which residue classes are available to prime terms.
What the Shift β Changes
For a fixed irrational α, changing β changes individual terms. Early prime counts, the first prime term, and local patterns can all move.
The leading asymptotic density does not change:
πB(x) ~ π(x) / α
for fixed β and irrational α > 1. The slope determines the main density; the shift determines where the selection window begins.
This distinction is especially important when comparing small numerical samples. Two sequences with the same irrational α but different β values can look noticeably different near the beginning while having the same limiting prime share.
The Golden Ratio Splits the Primes Between Two Beatty Sequences
Beatty sequences have a special complementary property. If two irrational numbers α and γ satisfy
1 / α + 1 / γ = 1,
then the homogeneous Beatty sequences {⌊αn⌋} and {⌊γn⌋} partition the positive integers: every positive integer occurs in exactly one of them.
The golden ratio gives the familiar pair
α = φ ≈ 1.6180339887
γ = φ² ≈ 2.6180339887
1 / φ + 1 / φ² = 1.
Since every positive integer belongs to exactly one of these two sequences, every prime also belongs to exactly one of them.
| Sequence | Integer Density | Asymptotic Share of Primes |
|---|---|---|
| Bφ | 1 / φ | ≈ 61.8034% |
| Bφ² | 1 / φ² | ≈ 38.1966% |
The two percentages add to 100%. This partition does not create new primes; it assigns every existing prime to one of two complementary irrational Beatty sequences.
Global Density Does Not Produce Regular Prime Spacing
The limit
πB(x) / π(x) → 1 / α
is a global counting statement. It does not say that every block of integers contains exactly the expected number of Beatty primes.
Local counts can be above or below their long-run prediction. Beatty primes can appear close together, and larger prime-free stretches can also occur. The same distinction appears in the ordinary distribution of primes: average density gives useful scale but does not identify the exact position of the next prime.
Global Density Versus Local Behavior
- A limiting share of 1 / α does not imply constant prime gaps.
- It does not force every finite interval to contain the expected number of primes.
- It does not make consecutive Beatty prime gaps converge to one fixed value.
- It does not identify whether a particular Beatty term is prime.
Prime Distribution in Shorter Intervals
Counting all Beatty primes up to x is only one form of the distribution problem. A harder question asks how many occur in a shorter interval from x to x + y.
For an irrational α > 1, asymptotic results are known when the interval length has the form y = xθ with θ > 5 / 8. In that range, the prime count has the expected leading form
y / (α log x).
For irrational slopes satisfying stronger Diophantine approximation conditions, the asymptotic range can be pushed further, including θ > 3 / 5 for slopes of finite type.
Sieve methods also give lower bounds in shorter ranges where a full asymptotic formula is harder to prove. This distinction matters: proving that an interval contains many Beatty primes is weaker than proving that their count matches the predicted main term.
Bounded Clusters of Beatty Primes
The irregular local spacing does not prevent Beatty primes from forming tight clusters infinitely often. Methods developed for small gaps between ordinary primes can be adapted to Beatty sequences, producing sets of several prime terms whose total diameter remains bounded under suitable conditions.
This does not mean that every pair of consecutive Beatty primes has a bounded gap. The statement is an existence result: sufficiently tight prime clusters recur even though other gaps may be much larger.
For two consecutive Beatty primes p and q, the quantity q − p is their prime gap only when no other prime belonging to that same Beatty sequence lies between them. Ordinary consecutive primes and consecutive primes within a Beatty subsequence are therefore different notions.
Where the Finite-Type Condition Enters
The basic density result for primes in an irrational Beatty sequence should not be confused with the stronger hypotheses used in some refined theorems.
An irrational number is described as being of finite type when its rational approximations cannot be exceptionally accurate too often. The technical definition belongs to Diophantine approximation, but its role here is straightforward: it gives tighter control over error terms arising from the irrational slope.
Basic Result and Stronger Results
Basic prime density: irrationality of α is enough for πB(x) ~ π(x) / α.
More uniform estimates: finite-type assumptions appear in results involving sharper short intervals, arithmetic progressions, and related error bounds.
Thus “α must be of finite type” should not be attached to every statement about Beatty primes. It is a condition used when the theorem asks for more control than the basic long-run density.
What the Asymptotic Formula Can and Cannot Predict
The formula
πB(x) ~ x / (α log x)
describes the leading growth of the prime count for an irrational Beatty sequence. It supports several precise interpretations while leaving individual primality questions unresolved.
- It predicts the long-run number of Beatty primes below a large cutoff.
- It predicts that the sequence captures a fraction 1 / α of all primes asymptotically.
- It does not give the exact number of Beatty primes below a finite cutoff.
- It does not determine the next Beatty prime.
- It does not make prime gaps regular.
- It does not apply unchanged to rational slopes.
- It does not prove that a particular term ⌊αn + β⌋ is prime; that remains an ordinary primality question for the resulting integer.