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Testing
Published in Zaven A. Karian, Edward J. Dudewicz, Modern Statistical, Systems, and GPSS Simulation, 2020
Zaven A. Karian, Edward J. Dudewicz
The first 8 of these were claimed to be prime by Father Martin Mersenne in 1644. Thus, 231 — 1, often used as m in random number generators, is a prime number. This fact was established by Euler in 1772. The larger numbers in this list are of much more recent date. For example, the three Mersenne numbers 2e – 1 for e=521,607,1279 were established as primes at the National Bureau of Standards (now the National Institute of Standards and Technology) in 1952. There are currently 32 known Mersenne primes, the largest of which has e=756,839, and was shown to be prime in 1992 by researchers at AEA Technology’s Harwell Laboratory in Britain. This number is, in fact, the largest known prime and has 227,832 digits.
Complexity and Cryptology
Published in J.-P. Barthelemy, G. Cohen, A. Lobstein, Catherine Fritsch-Mignotte, Maurice Mignotte, Algorithmic Complexity and communication problems, 2020
J.-P. Barthelemy, G. Cohen, A. Lobstein
A usual size for n is of 512 bits. Then the prime numbers p and q must be chosen around 2256 ≈ 1077. We know the distribution of prime numbers well enough (see Riesel, 1985) to estimate that between 1077 − 1070 and 1077 + 1070 for example (an interval of size 2 × 1070), there are approximately 2 • 1070/ ln(1077) prime numbers (In is the natural logarithm), that is approximately 2 • 1070/180. We have approximately a one in ninety chance of getting a prime number by taking an odd number at random in this interval. These figures ensure that the space of possible secret keys p and q is too large for an exhaustive search to be made, and that after picking several numbers at random, the probability of picking a prime number is not negligible. Verifying that a number of 256 bits is (pseudo-)prime can be done rapidly (cf. §3.1).
From Byzantine Consensus to Blockchain Consensus
Published in Kuan-Ching Li, Xiaofeng Chen, Hai Jiang, Elisa Bertino, Essentials of Blockchain Technology, 2019
There is a line or research on useful puzzles, i.e., on alternatives to PoW that spend energy solving useful tasks. The challenge is to find puzzles that are hard to compute, but with solutions that that can be verified efficiently. Primecoin seems to be the first cryptocurrency based on such an puzzle [145]. Its puzzles involve finding chains of prime numbers that are large enough to be hard to find, but not to verify. These prime numbers might be useful as cryptographic keys. Permacoin substitutes PoW by proofs-of-retrievability (PoR) to support distributed storage of data [146]. The purpose of a PoR is to certify that a node is using storage space to store a file. Therefore, Permacoin is a peer-to-peer file storage system in which nodes have an incentive to provide storage space, instead of mere altruism.
Meaning making in a sixth-grade mathematics classroom through touch screen technology
Published in International Journal of Mathematical Education in Science and Technology, 2022
This study builds on socio-cultural learning theories that argue for the individual being an active participant in his or her own education (Cobb & Bowers, 1999). Mathematics is often perceived as an abstract knowledge area, and the learning of mathematical concepts occurs through abstraction processes. In the natural world, learning the properties of a tangible object can often help us develop an understanding of that object. In mathematics, one can consider a ‘mathematical object’ (Sfard, 2008) as an abstract concept in which its content is, in a sense, hidden (Steinbring, 2005). To represent how mathematical concepts are abstract constructs (Thompson & Sfard, 1994), we can consider the case of prime numbers, which are one set of natural numbers. One can identify whether a natural number is prime by verifying whether the number satisfies the definition of a prime number. The fundamental theorem of arithmetic states that all natural numbers can be written uniquely as products of prime factors (Rosen, 1988). This fact expresses a multiplicative relationship that is initially hidden. Therefore, the presentation of mathematical objects through character signs and symbols is initially not directly related to its contents (Duval, 2017)
Fostering collateral creativity through teaching school mathematics with technology: what do teachers need to know?
Published in International Journal of Mathematical Education in Science and Technology, 2022
What is a prime number? One of the definitions is as follows: Prime number is a natural number greater than 1 which is not a product of smaller natural numbers. How can one find such numbers? The first number satisfying the definition is 2. It is a product of two numbers, 1 and 2, but only 1 is smaller than 2. The same can be said about 3. It is a product of two numbers 1 and 3, but only 1 is smaller than 3. Yet the number 4 is not a prime as it can be written as a product of two 2’s. In the 3rd century B.C., a Greek scholar Eratosthenes devised a method (nowadays called the sieve of Eratosthenes; the term sieve used in the previous section was borrowed from this context) of separating primes from natural numbers by continuously eliminating all multiples of 2, then all multiples of 3, then all multiples of 5 (the third prime number), then all multiples of 7 (as 7 is the first number to survive elimination by 2, 3, and 5), and so on.
Practical online assessment of mathematical proof
Published in International Journal of Mathematical Education in Science and Technology, 2022
Robert Thomas Bickerton, Chris J Sangwin
Mathematical theorems can be divided into two classes: specific and general. Specific theorems concern one object, or a unique situation. For example, the following classic proofs are all specific: (i) there are infinitely many prime numbers, (ii) the real numbers are uncountable and (iii) . General theorems have hypotheses which a range of examples do/do not satisfy, e.g.