Abstract
The sequence \(3, 5, 9, 11, 15, 19, 21, 25, 29, 35,\ldots \) consists of odd legs in right triangles with integer side lengths and prime hypotenuse. We show that the upper density of this sequence is zero, with logarithmic decay. The same estimate holds for the sequence of even legs in such triangles. We expect our upper bound, which involves the Erdős–Ford–Tenenbaum constant, to be sharp up to a double-logarithmic factor. We also provide a nontrivial lower bound. Our techniques involve sieve methods, the distribution of Gaussian primes in narrow sectors, and the Hardy–Ramanujan inequality.
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1 Background
The sequence OEIS A281505 concerns odd legs in right triangles with integer side lengths and prime hypotenuse. By the parametrisation of Pythagorean triples, these are positive integers of the form \(x^2 - y^2\), where \(x,y \in \mathbb N\) and \(x^2 + y^2\) is prime. Even legs are those of the form 2xy, where \(x, y \in \mathbb N\) and \(x^2 + y^2\) is an odd prime. Let \(\mathcal A\) be the set of odd legs, and \(\mathcal B\) the set of even legs that occur in such triangles. Consider the quantities
as \(N \rightarrow \infty \).
Let \(\mathcal P\) denote the set of primes. By a change of variables, observe that
Additionally, note that
where
We estimate \(\mathcal C(N)\), which is equivalent to estimating \(\mathcal B(N)\) and similar to estimating \(\mathcal A(N)\).
Let
be the Erdős–Ford–Tenenbaum constant. This constant is related to the number of distinct products in the multiplication table, and also arises in other contexts, for example, see [3, 4, 11, 12].
Theorem 1.1
We have
Since every prime \(p\equiv 1\pmod 4\) is representable as \(a^2+b^2\) with a, b integral, we have \(\mathcal C(N)\) unbounded. In fact, using the maximal order of the divisor function, we have \(\mathcal C(N) \geqslant N^{1-o(1)}\) as \(N\rightarrow \infty \). We obtain a strengthening of this lower bound.
Theorem 1.2
We have, as \(N\rightarrow \infty \),
Note that \(\log 4-1\approx 0.386\). Since \(\mathcal B(2N) = \mathcal C(N)\), we obtain the same bounds for \(\mathcal B(N)\). By essentially the same proofs, one can also deduce these bounds for \(\mathcal A(N)\).
To motivate the outcome, consider the following heuristic. There are typically \(\approx (\log n)^{\log 2}\) divisors of n, which follows from the normal number of prime factors of n, a result of Hardy and Ramanujan [8]. Moreover, given a factorisation \(n=ab\), the “probability” of \(a^2+b^2\) being prime is roughly \((\log n)^{-1}\). Since \(\log 2 < 1\), we expect the proportion to decay logarithmically. In the presence of biases and competing heuristics, this prima facie prediction should be taken with a few grains of salt. We use Brun’s sieve and the Hardy–Ramanujan inequality to formally establish our bounds. In addition, for Theorem 1.2 we use a result of Harman and Lewis [9] on the distribution of Gaussian primes in narrow sectors of the complex plane.
We write for the set of primes. We use Vinogradov and Bachmann–Landau notation. As usual, we write for the number of distinct prime divisors of n, and for the number of prime divisors of n counted with multiplicity. The symbols p and \(\ell \) are reserved for primes, and N denotes a large positive real number.
2 An upper bound
In this section, we establish Theorem 1.1. The Hardy–Ramanujan inequality [8] states that there exists a positive constant \(c_0\) such that uniformly for \(i \in \mathbb N\) and \(N\geqslant 3\) we have
By Mertens’s theorem and the fact that the sum of the reciprocals of prime powers higher than the first power converges, there is a positive constant \(c_1\) such that
Let \({\alpha }\) be a parameter in the range \(1< {\alpha }< 2\), to be specified in due course. We begin by bounding the size of the exceptional set
where
By the Hardy–Ramanujan inequality, we have
where \(k= \log \log N\), and therefore
Note that we have used here the elementary inequality \(1/L!<(e/L)^L\), which holds for all positive integers L and follows instantly from the Taylor series for \(e^L\). Thus,
For an integer \(n\geqslant 2\), write \(P^+(n)\) for the largest prime factor of n, and let \(P^+(1)=1\). By de Bruijn [1, Eq. (1.6)] we may bound the size of the exceptional set
by \(N/(\log N)^2\) for all sufficiently large numbers N. (Actually, the denominator may be taken as any fixed power of \(\log N\)).
Next, we estimate
For n counted by \(\mathcal C^*(N)\), we see by symmetry that we have \(n = ab_0 \ell \) for some \(a,b_0, \ell \in \mathbb N\) with \(\ell > N^{1/\log \log N}\) prime and \(a^2 + b_0^2 \ell ^2\) prime. Thus
where
We turn our attention to \(S(a,b_0)\). We may assume that \(ab_0\) is even and \(\gcd (a,b_0) = 1\), for otherwise \(S(a,b_0)= 0\). Observe that
where
To bound this from above, we apply Brun’s sieve [6, Corollary 6.2] with
and with the completely multiplicative density function g defined by
For this to be valid, we need to check that
where
We begin by noting that if \(p \in \mathcal P\) then the congruence
has g(p)p solutions \(m {\,\,{\mathrm{mod}}\,\,p}\). Observe that any divisor d of P(z) must be squarefree; thus, by the Chinese remainder theorem, the congruence
has g(d)d solutions \(m {\,\,{\mathrm{mod}}\,\,d}\). By periodicity, we now have
where \(M = X - d \lfloor X/d \rfloor \). This confirms (2.5), since \(0 \leqslant M < d\) and \(0 < g(d) \leqslant 1\).
We also need to check that
where \(V(z) = \prod _{p < z} (1- g(p))\), and where
This follows from the inequalities
Now [6, Corollary 6.2] tells us that
Remark 2.1
Note that we might equally well have used the version of Brun’s sieve from [7, p. 68], which is less precise, but somewhat easier to utilise. In fact, as kindly suggested by one of the referees, one could accomplish the same result using Brun’s pure sieve [6, Eq. (6.1)], which is nothing more than a strategic truncation of the inclusion-exclusion principle.
Substituting this into (2.4) yields
where
It follows from the multinomial theorem that
Letting \(m=j+k\), the binomial theorem now gives
where \(c_1\) is as in (2.1). In view of (2.2), we now have
Substituting this into (2.6) yields
By (2.3), our estimate for \(\#\mathcal E_2\), and (2.7), we have
where
We now choose \(1< {\alpha }< 2\) so as to maximise \(\mathcal M\). One might guess that this \({\alpha }\) solves
and indeed \({\alpha }= (\log 2)^{-1}\) does maximise \(\mathcal M\) on the interval (1, 2). With this choice of \({\alpha }\), we have
completing the proof of Theorem 1.1.
3 A lower bound
In this section, we establish Theorem 1.2. Let
Writing \(P^+(n)\) for the largest prime factor of \(n>1\), and \(P^+(1) = 1\), put
Let \(\varepsilon \) be a small positive real number, and let
Finally, write
As we seek a lower bound, we are free to discard some inconvenient elements of \(\mathcal C(N)\). Thus, by the Cauchy–Schwarz inequality, we have
where \(\mathcal S(N)\) is the number of quadruples \((a,b,c,d) \in \mathbb N^4\) such that
We first show that
For this, we use existing work counting Gaussian primes in narrow sectors. For convenience, we state the relevant result [9, Theorem 2].
Theorem 3.1
(Harman–Lewis) Let X be a large positive real number, and let \({\beta }, {\gamma }\) be real numbers in the ranges
Then
The implied constant is absolute.
Remark 3.2
The problem of counting Gaussian primes in narrow sectors has received quite some attention over the years, and still it is far from resolved. Rather than using Theorem 3.1 by Harman and Lewis [9], we could have used a weaker result by Kubilius [10] from the 1950s. We refer the interested reader to the introduction of [2] for more about the earlier history of this problem.
For positive integers \(i \leqslant \frac{ \log N}{10 \log 2}\), we apply this with
By Jordan’s inequality
observe that if \(a,b \in \mathbb N\), \(a^2 + b^2 \leqslant X\) and \({\theta }= \arctan (b/a) \leqslant \pi 2^{-i}\) then
Thus
confirming (3.2).
Next, we show that \(\# \mathcal L_j = o(N)\) (\(j=1,2,3\)).
Lemma 3.3
We have \(\# \mathcal L_1 = o(N)\).
Proof
By de Bruijn [1, Eq. (1.6)], we have
Thus, by symmetry, we have \(\# \mathcal L_1 \ll \frac{N}{\log N}\). \(\square \)
Lemma 3.4
We have
Proof
As \(\# \mathcal L_2 = \# \mathcal L_3\), we need only show this for \(j=2\). Taking out a prime factor \(\ell > N^{1/\log \log N}\) of ab, we have
where
As in the previous section, Brun’s sieve implies that
Therefore
where
As in the prior section, the multinomial theorem implies that
Since \((1+\varepsilon )(1-\log (1+\varepsilon ))<1\), using this estimate in (3.3) completes the proof of the lemma. \(\square \)
Combining (3.2) with Lemmas 3.3 and 3.4 gives
Lemma 3.5
If \(c' > \log 4 -1\) then
Proof
One component of the count is when \((a,b)=(c,d)\). This is the diagonal case, and it is easily estimated. By the sieve, the number of pairs \((a,b)\in \mathcal L\) with \(a\leqslant b\) is at most
which is negligible (note that this estimate shows that (3.5) is essentially tight).
For the nondiagonal case we imitate Sect. 2. If (a, b, c, d) is counted by \(\mathcal S(N)\), put
so that
Recall (3.4), and let \(\mathcal G\) be the set of \((g,u,v,w_0) \in \mathbb N^4\) such that
As \(P^+(ab) > N^{1/ \log \log N}\), we see by symmetry that
where
The fact that \(u \ne v\) ensures that there are three primality conditions defining \(S(g,u,v,w_0)\). To bound \(S(g,u,v,w_0)\) from above, we may assume without loss that \(guvw_0\) is even, and that the variables \(g,u,v,w_0\) are pairwise coprime, for otherwise \(S(g,u,v,w_0) = 0\). Paralleling Sect. 2, an application of Brun’s sieve reveals that
Substituting (3.7) into (3.6) yields
where
and T is as in (3.4). With \(U = 2T\), it follows from the multinomial theorem that
and a further application of the multinomial theorem gives
As \(U = 2(1+\varepsilon )\log \log N+O(1)\), we now have
Substituting this into (3.8) yields
As \(c' > \log 4 -1\), we may choose \(\varepsilon > 0\) to give \(\mathcal S(N) \ll _{c'} N (\log N)^{c'}\). \(\square \)
Combining (3.1) and (3.5) with Lemma 3.5 establishes Theorem 1.2.
4 A final comment
We conjecture that Theorem 1.1 holds with equality. For a lower bound, one might restrict attention to those pairs (a, b) with \(\omega (a)\approx \omega (b)\approx \frac{1}{2\log 2}\log \log N\). The upper bound for the second moment is analysed as in the paper, getting \(N/(\log N)^{\eta +o(1)}\); we expect that a more refined analysis would give
here. The difficulty is in obtaining this same estimate as a lower bound for the first moment. This would follow if we had an analogue of Theorem 3.1 in which a, b have a restricted number of prime factors. Such a result holds for the general distribution of Gaussian primes, at least if one restricts only one of a, b, see [5].
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Author's contributions
SC and CP jointly proved the theorems, drafted the manuscript, and polished it. Both authors have read and approved the final manuscript
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Acknowledgements
The authors were supported by the National Science Foundation under Grant No. DMS-1440140 while in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the Spring 2017 semester. The authors thank John Friedlander, Roger Heath-Brown, Zeev Rudnick, Andrzej Schinzel and the anonymous referees for helpful comments, and Tomasz Ordowski for suggesting the problem.
Dedication
This year (2017) is the 100th anniversary of the publication of the paper On the normal number of prime factors of a number n, by Hardy and Ramanujan, see [8]. Though not presented in such terms, their paper ushered in the subject of probabilistic number theory. Simpler proofs have been found, but the original proof contains a very useful inequality, one which we are happy to use yet again. We dedicate this note to that seminal paper.
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The authors declare that they have no competing interests.
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
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Chow, S., Pomerance, C. Triangles with prime hypotenuse. Res. number theory 3, 21 (2017). https://doi.org/10.1007/s40993-017-0086-6
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DOI: https://doi.org/10.1007/s40993-017-0086-6