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In a previous escapade we gave a collection of continued fractions involving Catalan's constant. This paper provides more general formulae governing those continued fractions. Having distinguished different cases associated to regions in the plan, we nickname those continued fractions \enquote{The Balkans} as they divide into areas which are related but still different in nature. Because we do not provide formal proofs of those machine-constructed formulae we do not claim them to be theorems. Still, each and every proposed formula was extensively tested numerically.

We provide the results of pattern recognition experiments on mathematical expressions. We give a few examples of conjectured results. None of which was thoroughly checked for novelty. We did not attempt to prove all the relations found and focused on their generation.

The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures involving even and odd $\zeta$ function values, logarithms etc. We show that many relations detected by the Ramanujan Machine Project stem from a specific algebraic observation and show how to generate infinitely many. This provides an automated proof and/or an explanation of many of the relations listed as conjectures by the project (although not all of them).

The Ramanujan Machine project detects new expressions related to constants of interest, such as $\zeta$ function values, $\gamma$ and algebraic numbers (to name a few). In particular the project lists a number of conjectures concerning the Catalan constant $G= 0.91596559\ldots$ We show how to generate infinitely many. We used an ad hoc software toolchain and rather tedious mathematical developments. Because we do not provide a proper peer-reviewed proof of the relations given here we do not claim them to be theorems.

Detection of rare traits or diseases in a large population is challenging. Pool testing allows covering larger swathes of population at a reduced cost, while simplifying logistics. However, testing precision decreases as it becomes unclear which member of a pool made the global test positive. In this paper we discuss testing strategies that provably approach best-possible strategy - optimal in the sense that no other strategy can give exact results with fewer tests. Our algorithms guarantee that they provide a complete and exact result for every individual, without exceeding $1/0.99$ times the number of tests the optimal strategy would require. This threshold is arbitrary: algorithms closer to the optimal bound can be described, however their complexity increases, making them less practical. Moreover, the way the algorithms process input samples leads to some individuals' status to be known sooner, thus allowing to take urgency into account when assigning individuals to tests.

As humanity struggles to contain the global COVID-19 pandemic, privacy concerns are emerging regarding confinement, tracing and testing. The scientific debate concerning privacy of the COVID-19 tracing efforts has been intense, especially focusing on the choice between centralised and decentralised tracing apps. The privacy concerns regarding COVID-19 testing, however, have not received as much attention even though the privacy at stake is arguably even higher. COVID-19 tests require the collection of samples. Those samples possibly contain viral material but inevitably also human DNA. Patient DNA is not necessary for the test but it is technically impossible to avoid collecting it. The unlawful preservation, or misuse, of such samples at a massive scale may hence disclose patient DNA information with far-reaching privacy consequences. Inspired by the cryptographic concept of "Indistinguishability under Chosen Plaintext Attack", this paper poses the blueprint of novel types of tests allowing to detect viral presence without leaving persisting traces of the patient's DNA. Authors are listed in alphabetical order., Comment: 10 pages, 1 figure

As humanity struggles to contain the global Covid-19 infection, prophylactic actions are grandly slowed down by the shortage of testing kits. Governments have taken several measures to work around this shortage: the FDA has become more liberal on the approval of Covid-19 tests in the US. In the UK emergency measures allowed to increase the daily number of locally produced test kits to 100,000. China has recently launched a massive test manufacturing program. However, all those efforts are very insufficient and many poor countries are still under threat. A popular method for reducing the number of tests consists in pooling samples, i.e. mixing patient samples and testing the mixed samples once. If all the samples are negative, pooling succeeds at a unitary cost. However, if a single sample is positive, failure does not indicate which patient is infected. This paper describes how to optimally detect infected patients in pools, i.e. using a minimal number of tests to precisely identify them, given the a priori probabilities that each of the patients is healthy. Those probabilities can be estimated using questionnaires, supervised machine learning or clinical examinations. The resulting algorithms, which can be interpreted as informed divide-and-conquer strategies, are non-intuitive and quite surprising. They are patent-free. Co-authors are listed in alphabetical order.