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The Piatetski-Shapiro sequences are sequences of the form $ (\left\lfloor {{n^c}} \right\rfloor)_{n = 1}^\infty $ for $ c > 1 $ and $ c \not\in \mathbb{N} $. It is conjectured that there are infinitely many primes in Piatetski-Shapiro sequences for $ c \in (1, 2) $. For every $ R \ge 1 $, we say that a natural number is an $ R $-almost prime if it has at most $ R $ prime factors, counted with multiplicity. In this paper, we prove that there are infinitely many $ R $-almost primes in Piatetski-Shapiro sequences if $ c \in (1, c_R) $ and $ c_R $ is an explicit constant depending on $ R $.

Modern beam shaping techniques have enabled the generation of optical fields displaying a wealth of structural features, which include three-dimensional topologies such as Möbius, ribbon strips and knots. However, unlike simpler types of structured light, the topological properties of these optical fields have hitherto remained more of a fundamental curiosity as opposed to a feature that can be applied in modern technologies. Due to their robustness against external perturbations, topological invariants in physical systems are increasingly being considered as a means to encode information. Hence, structured light with topological properties could potentially be used for such purposes. Here, we introduce the experimental realization of structures known as framed knots within optical polarization fields. We further develop a protocol in which the topological properties of framed knots are used in conjunction with prime factorization to encode information., Beam shaping methods can generate optical fields with nontrivial topologies, which are invariant against perturbations and thus interesting for information encoding. Here, the authors introduce the realization of framed optical knots to encode programs with the conjoined use of prime factorization.

There have been several efforts to apply quantum SAT solving methods to factor large integers. While these methods may provide insight into quantum SAT solving, to date they have not led to a convincing path to integer factorization that is competitive with the best known classical method, the Number Field Sieve. Many of the techniques tried involved directly encoding multiplication to SAT or an equivalent NP-hard problem and looking for satisfying assignments of the variables representing the prime factors. The main challenge in these cases is that, to compete with the Number Field Sieve, the quantum SAT solver would need to be superpolynomially faster than classical SAT solvers. In this paper the use of SAT solvers is restricted to a smaller task related to factoring: finding smooth numbers, which is an essential step of the Number Field Sieve. We present a SAT circuit that can be given to quantum SAT solvers such as annealers in order to perform this step of factoring. If quantum SAT solvers achieve any asymptotic speedup over classical brute-force search for smooth numbers, then our factoring algorithm is faster than the classical NFS.

Mathematics has an important role in the world of education because it is very supportive of knowledge in other fields, such as physics, chemistry, biology and science. However, most people consider mathematics to be very difficult and boring because it is abstract, so mathematics learning should be carried out in a fun and interesting manner and easily understood by students. Determining the KPK and FPB of an integer is one of the mathematical material that is often considered difficult by students in primary school, especially if the number is large enough. Usually the teacher uses a factor tree or prime factorization. It was never applied using props, so this article discusses the alternative methods of determining the KPK and FPB of an integer using the props. The visual aids used were KPK and FPB boxes. Through this teaching aid, the learning becomes more fun and students become more enthusiastic about learning and understanding the concept of material becomes more increased .

We introduceadmissible collectionsfor a finite group 𝐺 and use them to prove that most of the finite classical groups in non-defining characteristic satisfy theQuillen dimension at 𝑝 property, a strong version of Quillen’s conjecture, at a given odd prime divisor 𝑝 of|G|\lvert G\rvert. Compared to the methods in [M. Aschbacher and S. D. Smith, On Quillen’s conjecture for the 𝑝-groups complex,Ann. of Math. (2)137(1993), 3, 473–529], our techniques are simpler.

A famous conjecture of Hardy and Littlewood claims the subadditivity of the prime counting function, namely that ?(x+y) ? ?(x)+?(y) holds for all integers x, y ? 2, where ?(x) is the number of primes not exceeding x. It is widely believed nowadays that this conjecture is not true since Hensley and Richards stunningly discovered an incompatibility with the prime k-tuples conjecture. Despite this drawback, here we generalize the subadditivity conjecture to subsets of prime numbers possessing a rich collection of preassigned structures. We show that subadditivity holds in this extended manner over certain ranges of the parameters which are wide enough to imply that it holds in an almost all sense. Under the prime k-tuples conjecture, very large values of convex combinations of the prime counting function are obtained infinitely often, thereby indicating a strong deviation of ?(x) from being convex, even in a localized form. Finally, a Tauberian type condition is given for subsets of prime numbers which in turn implies an extension of a classical phenomenon, originally suggested by Legendre, about the asymptotically best fit functions to ?(x) of the shape x/(log x ? A)., NA

Let X be a smooth, complex Fano 4-fold, and rho(X) its Picard number. If X contains a prime divisor D with rho(X)-rho(D)>2, then either X is a product of del Pezzo surfaces, or rho(X)=5 or 6. In this setting, we completely classify the case where rho(X)=5; there are 6 families, among which one is new. We also deduce the classification of Fano 4-folds with rho(X)>4 with an elementary divisorial contraction sending a divisor to a curve., 16 pages

International audience; We establish new Bombieri-Vinogradov type estimates for a wide class of multiplicative arithmetic functions and derive several applications, including: a new proof of a recent estimate by Drappeau and Topacogullari for arithmetical correlations; a theorem of Erdős-Wintner type with support equal to the level set of an additive function at shifted argument; and a law of iterated logarithm for the distribution of prime factors of integers weighted by τ (n − 1) where τ denotes the divisor function.

In mathematics, the Riemann Hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part $\frac{1}{2}$. It is one of the seven Millennium Prize Problems selected by the Clay Mathematics Institute to carry a US 1,000,000 prize for the first correct solution. In 1915, Ramanujan proved that under the assumption of the Riemann Hypothesis, the inequality $\sigma(n) < e^{\gamma } \times n \times \log \log n$ holds for all sufficiently large $n$, where $\sigma(n)$ is the sum-of-divisors function and $\gamma \approx 0.57721$ is the Euler-Mascheroni constant. In 1984, Guy Robin proved that the inequality is true for all $n > 5040$ if and only if the Riemann Hypothesis is true. Let $n > 5040$ be $n = r \times q$, where $q$ denotes the largest prime factor of $n$. If $n > 5040$ is the smallest number such that Robin inequality does not hold, then we show the following inequality is also satisfied: $\sqrt[q]{e} + \frac{\log \log r}{\log \log n} > 2$.

The public parameters of the RSA cryptosystem are represented by the pair of integers N and e . In this work, first we show that if e satisfies the Diophantine equation of the form e x 2 − ϕ ( N ) y 2 = z for appropriate values of x , y and z under certain specified conditions, then one is able to factor N . That is, the unknown y x can be found amongst the convergents of e N via continued fractions algorithm. Consequently, Coppersmith’s theorem is applied to solve for prime factors p and q in polynomial time. We also report a second weakness that enabled us to factor k instances of RSA moduli simultaneously from the given ( N i , e i ) for i = 1 , 2 , ⋯ , k and a fixed x that fulfills the Diophantine equation e i x 2 − y i 2 ϕ ( N i ) = z i . This weakness was identified by solving the simultaneous Diophantine approximations using the lattice basis reduction technique. We note that this work extends the bound of insecure RSA decryption exponents.

Fermat’s Factoring Algorithm (FFA) is an integer factorisation methods factoring the modulus N using exhaustive search. The appearance of the Estimated Prime Factor (EPF) method reduces the cost of FFA’s loop count. However, the EPF does not work for balanced primes. This paper proposed the modified Fermat’s Factoring Algorithm 1-Estimated Prime Factor (mFFA1-EPF) that improves the EPF method. The algorithm works for factoring a modulus with unbalanced and balanced primes, respectively. The main results of mFFA1-EPF focused on three criteria: (i) the approach to select good candidates from a list of convergent continued fraction, (ii) the establishment of new potential initial values based on EPF, and (iii) the application of the above modification upon FFA. The resulting study shows the significant improvement that reduces the loop count of FFA1 via (improved) EPF compared to existing methods. The proposed algorithm can be executed without failure and caters for both the modulus N with unbalanced and balanced primes factor. The algorithm works for factoring a modulus with unbalanced and balanced primes.

We prove the irreducibility of integer polynomials $f(X)$ whose roots lie inside an Apollonius circle associated to two points on the real axis with integer abscisae $a$ and $b$, with ratio of the distances to these points depending on the canonical decomposition of $f(a)$ and $f(b)$. In particular, we obtain irreducibility criteria for the case where $f(a)$ and $f(b)$ have few prime factors, and $f$ is either an Enestr\"om-Kakeya polynomial, or has a large leading coefficient. Analogous results are also provided for multivariate polynomials over arbitrary fields, in a non-Archimedean setting., Comment: 21 pages, 1 figure

We prove a number of results regarding odd values of the Ramanujan $$\tau $$ -function. For example, we prove the existence of an effectively computable positive constant $$\kappa $$ such that if $$\tau (n)$$ is odd and $$n \ge 25$$ then either $$\begin{aligned} P(\tau (n)) \; > \; \kappa \cdot \frac{\log \log \log {n}}{\log \log \log \log {n}} \end{aligned}$$ or there exists a prime $$p \mid n$$ with $$\tau (p)=0$$ . Here P(m) denotes the largest prime factor of m. We also solve the equation $$\tau (n)=\pm 3^{b_1} 5^{b_2} 7^{b_3} 11^{b_4}$$ and the equations $$\tau (n)=\pm q^b$$ where $$3\le q < 100$$ is prime and the exponents are arbitrary nonnegative integers. We make use of a variety of methods, including the Primitive Divisor Theorem of Bilu, Hanrot and Voutier, bounds for solutions to Thue–Mahler equations due to Bugeaud and Győry, and the modular approach via Galois representations of Frey–Hellegouarch elliptic curves.

Let &sigma, ={&sigma, i:i&isin, I} be a partition of the set P of all prime numbers and let G be a finite group. We say that G is &sigma, primary if all the prime factors of |G| belong to the same member of &sigma, G is said to be &sigma, soluble if every chief factor of G is &sigma, primary, and G is &sigma, nilpotent if it is a direct product of &sigma, primary groups. It is known that G has a largest normal &sigma, nilpotent subgroup which is denoted by F&sigma, (G). Let n be a non-negative integer. The n-term of the &sigma, Fitting series of G is defined inductively by F0(G)=1, and Fn+1(G)/Fn(G)=F&sigma, (G/Fn(G)). If G is &sigma, soluble, there exists a smallest n such that Fn(G)=G. This number n is called the &sigma, nilpotent length of G and it is denoted by l&sigma, (G). If F is a subgroup-closed saturated formation, we define the &sigma, F-lengthn&sigma, (G,F) of G as the &sigma, nilpotent length of the F-residual GF of G. The main result of the paper shows that if A is a maximal subgroup of G and G is a &sigma, soluble, then n&sigma, (A,F)=n&sigma, (G,F)&minus, i for some i&isin, {0,1,2}.

In part I of this paper we studied additive decomposability of the set F y of the y -smooth numbers and the multiplicative decomposability of the shifted set G y = F y + { 1 } . In this paper, focusing on the case of ’large’ functions y , we continue the study of these problems. Further, we also investigate a problem related to the m-decomposability of k -term sumsets, for arbitrary k .

The importance of keeping information secret cannot be overemphasized especially in today,s digital world where eavesdroppers are rampant in our chanels of communication. This made the use of strong encryption schemes inevitable in order to safeguard the security of our system. RSA cryptosystem and its variants have been designed to provide confidentiality and integrity of data in our medium of communication. This paper reports new short decryption exponent attack on prime power with modulus $N=p^rq$ for $r\geq 2$ using continued fraction method which makes it vulnerable to Diophantine attack and breaks the security of the cryptosystem by factoring the modulus into its prime factors since the hardness relies on the integer factorization problem. The paper also shows that if the short decryption exponent $d

In this paper, we show an analogue of Kural, McDonald and Sah's result on Alladi's formula for global function fields. Explicitly, we show that for a global function field $K$, if a set $S$ of prime divisors has a natural density $\delta(S)$ within prime divisors, then $$-\lim_{n\to\infty} \sum_{\substack{1\le \deg D\le n\\ D\in \mathfrak{D}(K,S)}}\frac{\mu(D)}{|D|}=\delta(S),$$ where $\mu(D)$ is the M\"{o}bius function on divisors and $\mathfrak{D}(K,S)$ is the set of all effective distinguishable divisors whose smallest prime factors are in $S$. As applications, we get the analogue of Dawsey's and Sweeting and Woo's results to the Chebotarev Density Theorem for function fields, and the analogue of Alladi's result to the Prime Polynomial Theorem for arithmetic progressions. We also display a connection between the M\"obius function and the Fourier coefficients of modular form associated to elliptic curves. The proof of our main theorem is similar to the approach in Kural et al.'s article., Comment: 17 pages. To appear in Finite Fields and Their Applications

This paper is concerned with factor left prime factorization problems for multivariate polynomial matrices without full row rank. We propose a necessary and sufficient condition for the existence of factor left prime factorizations of a class of multivariate polynomial matrices, and then design an algorithm to compute all factor left prime factorizations if they exist. We implement the algorithm on the computer algebra system Maple, and two examples are given to illustrate the effectiveness of the algorithm. The results presented in this paper are also true for the existence of factor right prime factorizations of multivariate polynomial matrices without full column rank.

Consider a random sequence of $N$ integers, each chosen uniformly and independently from the set $\{1,\dots,x\}$. Motivated by applications to factorisation algorithms such as Dixon's algorithm, the quadratic sieve, and the number field sieve, Pomerance in 1994 posed the following problem: how large should $N$ be so that, with high probability, this sequence contains a subsequence, the product of whose elements is a perfect square? Pomerance determined asymptotically the logarithm of the threshold for this event, and conjectured that it in fact exhibits a sharp threshold in $N$. More recently, Croot, Granville, Pemantle and Tetali determined the threshold up to a factor of $4/\pi + o(1)$ as $x \to \infty$, and made a conjecture regarding the location of the sharp threshold. In this paper we prove both of these conjectures, by determining the sharp threshold for making squares. Our proof combines techniques from combinatorics, probability and analytic number theory; in particular, we use the so-called method of self-correcting martingales in order to control the size of the 2-core of the random hypergraph that encodes the prime factors of our random numbers. Our method also gives a new (and completely different) proof of the upper bound in the main theorem of Croot, Granville, Pemantle and Tetali., Comment: 83 pages

Let $\ell$ be a prime divisor of the order of a finite unitary reflection group. We classify up to conjugacy the parabolic and reflection subgroups that are minimal with respect to inclusion, subject to containing an $\ell$-Sylow subgroup. The classification assists in describing the $\ell$-Sylow subgroups of unitary reflection groups up to group isomorphism. This classification also relates to the modular representation theory of finite groups of Lie type. We observe that unless a parabolic subgroup minimally containing an $\ell$-Sylow subgroup is $G$ itself, the reflection subgroup within the parabolic minimally containing an $\ell$-Sylow subgroup is the whole parabolic subgroup., Comment: 14 pages, 6 tables

The security of RSA relies on the computationally challenging factorization of RSA modulus with being a large semi-prime consisting of two primes for the generation of RSA keys in commonly adopted cryptosystems. The property of both congruent to 1 mod 4, is used in Euler&rsquo, s factorization method to theoretically factorize them. While this caters to only a quarter of the possible combinations of primes, the rest of the combinations congruent to 3 mod 4 can be found by extending the method using Gaussian primes. However, based on Pythagorean primes that are applied in RSA, the semi-prime has only two sums of two squares in the range of possible squares . As becomes large, the probability of finding the two sums of two squares becomes computationally intractable in the practical world. In this paper, we apply Pythagorean primes to explore how the number of sums of two squares in the search field can be increased thereby increasing the likelihood that a sum of two squares can be found. Once two such sums of squares are found, even though many may exist, we show that it is sufficient to only find two solutions to factorize the original semi-prime. We present the algorithm showing the simplicity of steps that use rudimentary arithmetic operations requiring minimal memory, with search cycle time being a factor for very large semi-primes, which can be contained. We demonstrate the correctness of our approach with practical illustrations for breaking RSA keys. Our enhanced factorization method is an improvement on our previous work with results compared to other factorization algorithms and continues to be an ongoing area of our research.

Let $$\theta $$ be an irrational number and $$\varphi $$ a real number. Let $$C > 2$$ and $$\varepsilon > 0$$ . There are infinitely many positive integers n free of prime factors $$> (\log n )^C$$ , such that $$\Vert \theta n + \varphi \Vert < n^{-\left( \frac{1}{3} - \frac{2}{3C}\right) + \varepsilon }$$ . Here $$\Vert y\Vert $$ is the distance from y to $$\mathbb Z$$ .

We build on the work of Drakakis et al. (2011) on the maximal cross-correlation of the families of Welch and Golomb Costas permutations. In particular, we settle some of their conjectures. More precisely, we prove two results. First, for a prime $p\ge 5$ , the maximal cross-correlation of the family of the $\varphi (p-1)$ different Welch Costas permutations of $\{1,\ldots,p-1\}$ is $(p-1)/t$ , where $t$ is the smallest prime divisor of $(p-1)/2$ if $p$ is not a safe prime and at most $1+p^{1/2}$ otherwise. Here $\varphi $ denotes Euler’s totient function and a prime $p$ is a safe prime if $(p-1)/2$ is also prime. Second, for a prime power $q\ge 4$ the maximal cross-correlation of a subfamily of Golomb Costas permutations of $\{1,\ldots,q-2\}$ is $(q-1)/t-1$ if $t$ is the smallest prime divisor of $(q-1)/2$ if $q$ is odd and of $q-1$ if $q$ is even provided that $(q-1)/2$ and $q-1$ are not prime, and at most $1+q^{1/2}$ otherwise. Note that we consider a smaller family than Drakakis et al. Our family is of size $\varphi (q-1)$ whereas there are $\varphi (q-1)^{2}$ different Golomb Costas permutations. The maximal cross-correlation of the larger family given in the tables of Drakakis et al. is larger than our bound (for the smaller family) for some $q$ .

Given disjoint subsets $T_1,\ldots,T_m$ of "not too large" primes up to $x$, we establish that for a random integer $n$ drawn from $[1,x]$, the $m$-dimensional vector enumerating the number of prime factors of $n$ from $T_1,\ldots,T_m$ converges to a vector of $m$ independent Poisson random variables. We give a specific rate of convergence using the Kubilius model of prime factors. We also show a universal upper bound of Poisson type when $T_1,\ldots,T_m$ are unrestricted, and apply this to the distribution of the number of prime factors from a set $T$ given that $n$ has $k$ total prime factors., v3. Improved the range of k in Theorem 4. Added several references to related work. To appear in the Math. Proc. Cambridge Phil. Soc

In spite of remarkable progress in machine learning techniques, the state-of-the-art machine learning algorithms often keep machines from real-time learning (online learning) due in part to computational complexity in parameter optimization. As an alternative, a learning algorithm to train a memory in real time is proposed, which is named as the Markov chain Hebbian learning algorithm. The algorithm pursues efficient memory use during training in that (i) the weight matrix has ternary elements (-1, 0, 1) and (ii) each update follows a Markov chain--the upcoming update does not need past weight memory. The algorithm was verified by two proof-of-concept tasks (handwritten digit recognition and multiplication table memorization) in which numbers were taken as symbols. Particularly, the latter bases multiplication arithmetic on memory, which may be analogous to humans' mental arithmetic. The memory-based multiplication arithmetic feasibly offers the basis of factorization, supporting novel insight into the arithmetic., Comment: 25 pages, 4 figures

Variational algorithm using Quantum Approximate Optimization Algorithm (QAOA) can solve the prime factorization problem in near-term noisy quantum computers. Conventional Variational Quantum Factoring (VQF) requires a large number of 2-qubit gates (especially for factoring a large number) resulting in deep circuits. The output quality of the deep quantum circuit is degraded due to errors limiting the computational power of quantum computing. In this paper, we explore various transformations to optimize the QAOA circuit for integer factorization. We propose two criteria to select the optimal quantum circuit that can improve the noise resiliency of VQF., 6 pages, 7 figures, submitted to ISLPED 2020 for review

Summary In the article we formalize some properties needed to prove that sequences of prime reciprocals are divergent. The aim is to show that the series exhibits log-log growth. We introduce some auxiliary notions as harmonic numbers, telescoping series, and prove some standard properties of logarithms and exponents absent in the Mizar Mathematical Library. At the end we proceed with square-free and square-containing parts of a natural number and reciprocals of corresponding products.

So far, the concept of the Greatest Common Divisor (GCD) and the Least Common Multiple (LCM) of two numbers are still being taught procedurally whether it is seeking multiples or by understanding prime numbers and prime factorization. This way often makes the students difficulties in determining the GCD and LCM. To overcome such things, the authors will offer a solution in learning mathematics material GCD and LCM in elementary school taught with Approach Realistic Mathematics Education (RME) food package presentation model. Mathematical learning with this approach is believed to make students learn as if they find themselves GCD and LCM concepts learned and will make learning more meaningful so that ultimately will improve students' understanding on GCD and LCM concepts.

This paper is intended as a sequel to a paper arXiv:0803.2636 written by four of the coauthors here. In the paper, they proved a stronger form of the Erd\H{o}s-Mirksy conjecture which states that there are infinitely many positive integers $x$ such that $d(x)=d(x+1)$ where $d(x)$ denotes the number of divisors of $x$. This conjecture was first proven by Heath-Brown in 1984, but the method did not reveal the nature of the set of values $d(x)$ for such $x$. In particular, one could not conclude that there was any particular value $A$ for which $d(x)=d(x+1)=A$ infinitely often. In the previous paper arXiv:0803.2636, the authors showed that there are infinitely many positive integers $x$ such that both $x$ and $x+1$ have exponent pattern $\{2,1,1,1\}$, so $d(x)=d(x+1)=24$. Similar results were known for certain shifts $n$, i.e., $x$ and $x+n$ have the same fixed exponent pattern infinitely often. This was done for shifts $n$ which are either even or not divisible by the product of a pair of twin primes. The goal of this paper is to give simple proofs of results on exponent patterns for an arbitrary shift $n$., Comment: 10 pages

We find asymptotics for $S_{K,c}(x)$, the number of positive integers below $x$ whose number of prime factors is $c \; \mathrm{mod}\; K$. We study this question in the context of Beurling integers., 5 pages

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the $j$-invariant of the elliptic curve is constant. In more detail, given an elliptic curve $E$ with a point $P$ of infinite order, the sequence $D_1$, $D_2, \ldots$ of denominators of multiples $P$, $2P,\ldots$ of $P$ is a strong divisibility sequence in the sense that $\gcd(D_m, D_n) = D_{\gcd(m,n)}$. This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences. A number $N$ is called a Zsigmondy bound of the sequence if each term $D_{n}$ with $n>N$ presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over $\mathbf{Q}$ is $30$ by Bilu-Hanrot-Voutier, 2000, but finding such a bound remains an open problem in genus one, both over $\mathbf{Q}$ and over function fields. We prove that the optimal Zsigmondy bound for ordinary elliptic divisibility sequences over function fields is $2$ if the $j$-invariant is constant. In the supersingular case, we give a complete classification of which terms can and cannot have a new prime factor., 24 pages

In 2009, Grant, Shallit, and Stoll constructed a large family of pseudorandom sequences, called generalized Rudin--Shapiro sequences, for which they established some results about the average of discrete correlation coefficients of order 2 in cases where the size of the alphabet is a prime number or a squarefree product of primes. We establish similar results for an even larger family of pseudorandom sequences, constructed via difference matrices, in the case of an alphabet of any size. The constructions generalize those from Grant et al. In the case where the size of the alphabet is squarefree and where there are at least two prime factors, we obtain an improvement in the error term by comparison with the result of Grant et al., Comment: 26 pages

We consider the number π(x, y; q, a) of primes p ⩽ x such that p ≡ a (mod q) and (p − a)/q is free of prime factors greater than y. Assuming a suitable form of Elliott-Halberstam conjecture, it is proved that π(x, y; q, a) is asymptotic to ρ(log(x/q)/log y)π(x)/φ(q) on average, subject to certain ranges of y and q, where ρ is the Dickman function. Moreover, unconditional upper bounds are also obtained via sieve methods. As a typical application, we may control more effectively the number of shifted primes with large prime factors.

21st IFAC World Congress on Automatic Control - Meeting Societal Challenges, IFAC, Berlin, Germany, 11 Jul 2020 - 17 Jul 2020; IFAC-PapersOnLine 53(2), 4316-4321 (2020). doi:10.1016/j.ifacol.2020.12.2487 special issue: "21th IFAC World Congress / Edited by Rolf Findeisen, Sandra Hirche, Klaus Janschek, Martin Mönnigman", Published by Elsevier, Frankfurt ; München [u.a.]

Connected sum and trivalent vertex sum are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the tunnel number of a composite genus 2 spatial graph. For the tunnel number of a composite Brunnian $\theta$-curve, our result implies that the tunnel number is at least the number of summands, as in the knot case. We also prove a version of a theorem of Morimoto for knots: the tunnel number of a composite m-small genus 2 spatial graph is at least the sum of the tunnel numbers of the factors. We also study lower bounds for the bridge number of composite genus 2 graphs. In particular, our results imply that for a Brunnian composite $\theta$-curve having $m$ factors in its prime factorization, the bridge number is at least $m+3/2$., Comment: 35 pages, 14 figures. Accepted by Pacific Journal of Mathematics