• Vivien Dracon
• GPT o3
• Gemini 2.5 Pro
• Sonnet 3.7 Sonnet

Prime Terms of the Binary-Hybrid Sequence ($s_n = 2^{2n+5} + 2^{n+2} + 1$): Half-Density, Proth-Type Testing, and Maximal-Period Reciprocals

Abstract: We investigate the sequence of integers defined by [ $s_n = 2^{2n+5} + 2^{n+2} + 1$ ] with emphasis on those terms $s_n$ that are prime. We show that this sequence produces several primes for specific values of $n$ (starting with $n=0,1,3,7,23,\dots$), and we analyze the remarkable property that for these prime terms $p=s_n$, the binary expansion of $1/p$ has maximal period in the sense that the repetend length equals Euler's totient $\varphi(p)$. Equivalently, $2$ is a primitive root modulo $p$. We provide a number-theoretic analysis of the sequence, proving structural constraints on $n$ that govern when $s_n$ can be prime and when $2$ can be a generator of $(\mathbb{Z}/s_n\mathbb{Z})^*$. In particular, we show that aside from the smallest case $p=37$, any prime in this sequence must lie in the congruence class $1\pmod{8}$ – a restriction that explains why $2$ has full multiplicative order in only special cases. We connect our results to Proth numbers, noting that $s_n$ has the form $k \cdot 2^m + 1$ and satisfies a Proth-like primality criterion. A detailed implementation for testing the primality of $s_n$ is described, combining modular arithmetic filters and Proth's theorem for an efficient search. We also discuss potential applications of these primes, including their use in cryptography and pseudorandom number generation due to their large order properties, and we examine how the condition of maximal period reciprocals can serve as a strong primality indicator. Finally, we summarize our findings and suggest avenues for further research, such as exploring the infinitude of primes in this sequence and generalizations to similar binary-form expressions.

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Full-Period Prime Reciprocals in a Modified Exponential Sequence

Abstract: We investigate the integer sequence defined by $t_n = 2^{2n+5} + 2^{n+2} + 3$ for $n\ge 0$, focusing on its prime values and the full-period property of their reciprocals in base 2. A prime $p=t_n$ has a full-period binary reciprocal if 2 is a primitive root modulo $p$, equivalently if the binary expansion of $1/p$ has maximal period $p-1$. We provide background on full-period reciprocals and primitive roots, then analyze number-theoretic properties of $t_n$ that influence its primality and the order of 2 modulo $t_n$. In particular, we show $t_n$ is always congruent to 3 mod 8, ensuring $2$ is a quadratic non-residue modulo $t_n$ (a necessary condition for a full period). We prove that $t_n$ is divisible by 3 for even $n$, restricting primes to odd $n$. We then discuss which odd $n$ yield primes and present empirical results: for $n=1,3,7,13,15,17$, the values $t_n$ are prime, and among these, $2$ is a primitive root for $n=1,3,17$ (full period reciprocals) while for $n=7,13,15$ the period is one-third of $p-1$. We formulate propositions that partially explain these observations. An efficient implementation for searching prime values of $t_n$ and verifying the primitive root property is described. Finally, we explore applications of such primes in cryptography (e.g. Diffie–Hellman with generator 2) and pseudorandom number generation, and consider directions for future research, including conjectures on the infinitude of primes in this sequence and generalizations to related forms.

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A Comparative Study of Primes and Periods in the Sequences $2^{2n+5}+2^{n+2}+1$ and $2^{2n+5}+2^{n+2}+3$

Abstract: We investigate two closely related integer sequences defined by $[s_n = 2^{2n+5} + 2^{n+2} + 1, \qquad t_n = 2^{2n+5} + 2^{n+2} + 3,]$ for integer $n$. Despite the only difference being the additive constant (1 vs. 3), these sequences exhibit markedly different number-theoretic behaviors. We compare their propensity to produce prime numbers and analyze for which values of $n$ each sequence yields primes. We then examine those prime outputs to determine when they have full-period binary reciprocals – that is, when 2 is a primitive root modulo the prime. We find that $t_n$ yields primes slightly less frequently than $s_n$ in initial ranges (with all $t_n$ primes occurring at odd $n$, whereas $s_n$ primes occur at both even and odd indices). Furthermore, every prime of the form $s_n$ falls into a single residue class modulo 8 (namely 1 mod 8), implying 2 is a quadratic residue and not a primitive root in those cases, whereas primes of the form $t_n$ are 3 mod 8, making 2 a quadratic non-residue and often a primitive root. We provide proofs of these residue properties and divisibility patterns (e.g. $3\mid t_n$ for even $n$, while $s_n$ avoids small prime divisors like 3 in all cases). An empirical study up to several large $n$ is included, with a table comparing values of $n$, the corresponding $s_n$ and $t_n$, their primality, $\varphi$-values, and the order of 2 modulo $s_n$ or $t_n$ (when prime). We interpret these results theoretically: the constant term differences lead to $s_n$ and $t_n$ living in different congruence classes (mod 8) and hence different allowed factorizations and multiplicative orders. Finally, we discuss implications for cryptographic and pseudorandom applications. Primes from the $t_n$ sequence can serve as full-period primes in base 2 (useful for maximum-length recurring sequences and cryptographic generators), whereas $s_n$ primes inherently have a limited period in base-2 expansions. We conclude with open questions, including the likelihood of infinitely many primes in either sequence and the density of indices yielding primes, connecting to broader conjectures like Artin’s primitive root conjecture and the distribution of prime values of polynomials.

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Exploring the Twin-Prime Phenomenon in Two Related Exponential Sequences

Abstract: We investigate a curious twin-prime phenomenon arising from two related exponential sequences of integers:

• $s_n = 2^{2n+5} + 2^{n+2} + 1$, and
• $t_n = 2^{2n+5} + 2^{n+2} + 3$,

for $n \ge 0$. These sequences produce prime numbers in tandem for certain small values of $n$ (notably $n=1,3,7$), yielding prime pairs $(s_n,,t_n)$ that differ by 2 – i.e. twin primes. We explore the structural reasons behind this phenomenon and assess its generality. In particular, we examine congruence properties showing that $s_n \equiv 1 \pmod{8}$ while $t_n \equiv 3 \pmod{8}$, and discuss how this forces the multiplicative order of 2 modulo $s_n$ (when prime) to be half of the full group order, whereas for $t_n$ (when prime) the order of 2 is often full (making 2 a primitive root modulo $t_n$). We present computational data up to $n=500$ demonstrating the rarity of twin primes in these sequences beyond the few initial cases. Using heuristics in the spirit of the Bateman–Horn conjecture, we argue that the probability of infinitely many such twin primes is exceedingly low (likely only finitely many exist). Connections to Artin’s conjecture on primitive roots are made to interpret the observed full-period reciprocals of $1/t_n$ in base-2. Finally, we discuss cryptographic implications of primes that admit 2 as a primitive root (maximal period primes), noting that the $t_n$ sequence provides a source of such primes (albeit sparse), which could be relevant for pseudorandom number generation and hash functions requiring long periodic sequences.