Brazilian Primes Which Are Also Sophie Germain Primes
Jon Grantham, Hester Graves

TL;DR
This paper disproves Schott's conjecture by identifying Brazilian primes that are also Sophie Germain primes, providing a comprehensive enumeration of all such counterexamples up to 10^46.
Contribution
It is the first to explicitly find and list all Brazilian primes that are also Sophie Germain primes up to 10^46, disproving the previous conjecture.
Findings
Counterexamples up to 10^46 identified
Schott's conjecture disproved
Complete enumeration of such primes provided
Abstract
We disprove a conjecture of Schott that no Brazilian primes are Sophie Germain primes. We enumerate all counterexamples up to .
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Taxonomy
TopicsAnalytic Number Theory Research · Limits and Structures in Graph Theory · Advanced Mathematical Identities
**BRAZILIAN PRIMES WHICH ARE ALSO SOPHIE GERMAIN PRIMES
Jon Grantham
**Institute for Defense Analyses/Center for Computing Sciences, Bowie, Maryland
**Hester Graves
**Institute for Defense Analyses/Center for Computing Sciences, Bowie, Maryland
Abstract
We disprove a conjecture of Schott that no Brazilian prime is a Sophie Germain prime. We compute all counterexamples up to . We prove conditional asymptotics for the number of Brazilian Sophie Germain primes up to .
1 Introduction
The term “Brazilian numbers” comes from the 1994 Iberoamerican Mathematical Olympiad [8] in Fortaleza, Brazil, in a problem proposed by the Mexican math team.111The term appears as “sensato” in the original problem [6]. The authors are puzzled by the discrepancy with [8].. They became a topic of lively discussion on the forum mathematiques.net. Bernard Schott [5] summarized the results in the standard reference on Brazilian numbers.
Definition 1**.**
A Brazilian number is an integer whose base- representation has all the same digits for some .**
In other words, is Brazilian if and only if , with . These numbers are A125134 in the Online Encylopedia of Integer Sequences (OEIS).
A Brazilian prime (or “prime repunit”) is a Brazilian number that is prime; by necessity, and . See A085104 in the OEIS for the sequence of Brazilian primes. In 2010, Schott [5] conjectured that no Brazilian prime is also a Sophie Germain prime.
Sophie Germain discovered her eponymous primes while trying to prove Fermat’s Last Theorem; her work was one of the first major steps towards a proof.
Definition 2**.**
A Sophie Germain prime is a prime such that is also prime.**
Germain showed that if is such a prime, then there are no non-zero integers , not divisible by , such that . If is a Sophie Germain prime, then we say that is a safe prime.
It is conjectured that there are infinitely many Sophie Germain primes, but the claim is still unproven. The Bateman-Horn conjecture [1] implies that the number of Sophie Germain primes less than is asymptotic to , where .
See [3] for further information about Sophie Germain primes.
2 Enumerating Counterexamples
To aid our search, we use a few lemmas.
Lemma 1**.**
If is a Brazilian prime, then is an odd prime.
Proof.
Recall is divisible by the th cyclotomic polynomial for ; therefore can only be prime if is also prime. Note that because , so is an odd prime. ∎
The preceding lemma is also Corollary 4.1 of Schott [5].
Lemma 2**.**
If is a Brazilian prime and a Sophie Germain prime, then (mod ) and (mod ).
Proof.
If is a Sophie Germain prime, then cannot divide the safe prime , so cannot be congruent to (mod ). The number is not Brazilian, so and thus (mod ).
If , then
[TABLE]
which is a contradiction. Lemma 1 states that is an odd prime, so if (mod , then (mod ), a contradiction. We conclude that (mod ), so that (mod ), and therefore (mod ). ∎
For , we use a modification of the technique described in [7] to compute a table of length- Brazilian primes up to . We will describe this computation in full in a forthcoming paper [2]. Of these, are Sophie Germain primes. The smallest is . We very easily prove the primality of Sophie German primes with the Pocklington-Lehmer test.
For , we very quickly enumerate all possibilities for . We find Brazilian Sophie Germain primes for , and none for larger . (We have .) The smallest is
[TABLE]
While we disprove Schott’s conjecture, we do have a related proposition.
Proposition 1**.**
The only Brazilian prime which is a safe prime is .
Proof.
If is a safe prime, then must also be prime. This expression, however, is divisible by , which is only prime when and . ∎
The list of Brazilian Sophie Germain primes is A306845 in the OEIS. The first few counterexamples were also discovered by Giovanni Resta and Michel Marcus; see the comments for A085104.
3 Conditional Results
The infinitude of Brazilian Sophie Germain primes, as well as the asymptotic number of them, is the consequence of well-known conjectures.
Proposition 2**.**
Assuming Schnizel’s Hypothesis H, there are infinitely many Brazilian Sophie Germain primes.
Proof.
Recall that Hypothesis H [4] says that any set of polynomials, whose product is not identically zero modulo any prime, is simultaneously prime infinitely often. Take our two polynomials to be and . Then is Brazilian and . Rather than checking congruences, it suffices to note the existence of the above primes of this form to see that the conditions of Hypothesis H are satisfied. ∎
The Bateman-Horn Conjecture [1] implies a more precise statement about the number of Brazilian Sophie Germain primes.
Proposition 3**.**
For an odd prime , let be the th cyclotomic polynomial. Assuming the Bateman-Horn Conjecture, the number of values of such that and are simultaneously prime is [math] or , for some positive constant , depending on whether is identically zero modulo some prime .
Proof.
This follows immediately from the Bateman-Horn conjecture, with , where is the number of roots of modulo . ∎
Corollary 1**.**
Assuming the Bateman-Horn Conjecture, the number of Brazilian Sophie Germain primes up to is , for some .
Proof.
To find the number of Brazilian Sophie Germain primes less than of the form for a fixed , we apply the preceding proposition, substituting , and get , with . We sum over all (mod ) and notice that the term dominates. We can thus take . ∎
**Acknowledgement. ** Thanks to Enrique Treviño for helping track down the reference in [6], and to the referee for helpful comments.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[1] P. T. Bateman and R. A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363–367.
- 2[2] J. Grantham and H. Graves, Goldbach variations, in preparation, 2020.
- 3[3] P. Ribenboim, The New Book of Prime Number Records , Springer-Verlag, New York, 1996.
- 4[4] A. Schinzel and W. Sierpiński, Sur certaines hypothèses concernant les nombres premiers, Acta Arith. 4 (1958), 185–208; erratum 5 (1958), 259.
- 5[5] B. Schott, Les nombres brésiliens, Quadrature (2010), 30–38.
- 6[6] E. Wagner, G. T. de Araujo Moreira Carlos, P. Fauring, F. Gutiérrez, and A. Wykowski, 10 Olimpíadas Iberoamericanas de Matemática , OEI, Madrid, 1996.
- 7[7] M. Wolf, Some conjectures on primes of the form m 2 + 1 superscript 𝑚 2 1 m^{2}+1 , J. Comb. Number Theory 5 (2013), 103–131.
- 8[8] R. E. Woodrow, The Olympiad corner, Crux Mathematicorum 24 (1998), 385–395.
