A new generalized definition of Mersenne numbers is proposed of the form an−a−1n, called global generalized Mersenne numbers and noted GMa,n with base a and exponent n positive integers. The ...properties are investigated for prime n and several theorems on Mersenne numbers regarding their congruence properties are generalized and demonstrated. It is found that for any a, GMa,n−1 is even and divisible by n, a and a−1 for any prime n>2, and by aa−1+1 for any prime n>5. The remaining factor is a function of triangular numbers of a−1, specific for each prime n. Four theorems on Mersenne numbers are generalized and four new theorems are demonstrated, showing first that GMa,n≡1or7mod12 depending on the congruence of amod4; second, that GMa,n−1 are divisible by 10 if n≡1mod4 and, if n≡3mod4, GMa,n≡1or7or9mod10, depending on the congruence of amod5; third, that all factors ci of GMa,n are of the form 2nfi+1 such that ci is either prime or the product of primes of the form 2nj+1, with fi,j natural integers; fourth, that for prime n>2, all GMa,n are periodically congruent to ±1or±3mod8 depending on the congruence of amod8; and fifth, that the factors of a composite GMa,n are of the form 2nfi+1 with fi≡umod4 with u=0, 1, 2 or 3 depending on the congruences of nmod4 and of amod8. The potential use of generalized Mersenne primes in cryptography is shortly addressed.
The classical problem of finding all integers a and M such that the sums of M consecutive squared integers a+i2 equal the squared integer s2, where M is the number of terms in the sum, a2 the first ...term and a≥1, 0≤i≤M−1, yields remarkable regular linear features when plotting the values of M as a function of a. These linear features correspond to groupings of pairs of a values for successive same values of M found on either side of straight lines of equation μM=2a+c, where c is an integer constant and μ a parameter taking some rational values, called allowed values. We find expressions of a and s as a function of M for the allowed values of μ and M and parametric expressions of a, M, and s. Further, Pell equations deduced from the conditions of M are solved to find the allowed values of μ and to provide all solutions in a and M. These results yield new insights into the overall properties of the classical problem of the sums of consecutive squared integers equal to squared integers and allow us to solve this problem completely by providing all solutions in infinite families.
We search for triangular numbers that are multiples of other triangular numbers. For any positive non-square integer multiplier, there is an infinity of multiples of triangular numbers that are ...triangular numbers and recurrent relations are deduced theoretically in function of two parameters. If the multiplier is a squared integer, there is either one or no solution, depending on the multiplier value.
Abstract
Long bone fractures are a concern in long-duration exploration missions (LDEM) where crew autonomy will exceed the current Low Earth Orbit paradigm. Current crew selection assumptions ...require extensive complete training and competency testing prior to flight for off-nominal situations. Analogue astronauts (n = 6) can be quickly trained to address a single fracture pattern and then competently perform the repair procedure. An easy-to-use external fixation (EZExFix) was employed to repair artificial tibial shaft fractures during an inhabited mission at the Mars Desert Research Station (Utah, USA). Bone repair safety zones were respected (23/24), participants achieved 79.2% repair success, and median completion time was 50.04 min. Just-in-time training in-mission was sufficient to become autonomous without pre-mission medical/surgical/mechanical education, regardless of learning conditions (p > 0.05). Similar techniques could be used in LDEM to increase astronauts’ autonomy in traumatic injury treatment and lower skill competency requirements used in crew selection.
Aircraft parabolic flights provide repetitively up to
20
s
of reduced gravity during ballistic flight manoeuvres. Parabolic flights are used to conduct short microgravity investigations in Physical ...and Life Sciences, to test instrumentation and to train astronauts before a space flight. The European Space Agency (ESA) has organized since 1984 thirty parabolic flight campaigns for microgravity research experiments utilizing six different airplanes. More than 360 experiments were successfully conducted during more than 2800 parabolas, representing a cumulated weightlessness time of
15
h
30
m
.
This paper presents the short duration microgravity research programme of ESA. The experiments conducted during these campaigns are summarized, and the different airplanes used by ESA are shortly presented. The technical capabilities of the Airbus A300 ‘Zero-G’ are addressed. Some Physical Science, Technology and Life Science experiments performed during the last ESA campaigns with the Airbus A300 are presented to show the interest of this unique microgravity research tool to complement, support and prepare orbital microgravity investigations.
It is shown that orbital period ratios of successive planets in the Solar System, of satellites in giant planet systems and of exoplanets in exoplanetary systems are preferentially closer to ...irreducible fractions formed with Fibonacci numbers between 1 and 8 than to other fractions, in a ratio of approximately 60% vs 40%. Furthermore, if sets of minor planets are chosen with gradually smaller inclinations and eccentricities, the proximity to Fibonacci fractions of their period ratios with Jupiter or Mars’ period tends to increase. Finally, a simple model explains why the resonance of the form
P
1
P
2
=
p
p
+
q
, with
P
1
and
P
2
orbital periods of planets or satellites and
p
and
q
small integers, are stronger and more commonly observed for
p
and
(
p
+
q
)
being both small Fibonacci numbers than for other cases.
Access to earthbound weightlessness is critical to many branches of applied sciences. Besides, several space systems require microgravity testing before their launch. Existing solutions (drop towers, ...parabolic flights, sounding rockets) offer variable durations and qualities of microgravity environment, but their cost and lead times make them unpractical for small actors such as universities or start-up companies. This leads to a growing interest for alternative microgravity platforms. Here, we study the use of gliders to perform parabolic flights at a lower cost, and we propose a systematic quantification of glider’s 0-g flight capabilities. Results of our flight test campaign show that gliders offer up to 5.5s of weightlessness, with excursions below 0.1g, and a satisfactory level of repeatability. Besides, the recordings do not suffer from the increased level of vibrations generated by piston engines, typical of light-aircraft-based alternatives. Operational considerations associated with glider parabolic flights are also discussed. Finally, we conclude that a microgravity platform based on gliders would be suitable especially for compact experiments and equipment in order to support accelerated design and development, or to produce preliminary experimental results.
In the first part of this paper, we considered several theoretical models, a static and four dynamic models without rebounds, of the throw of a fair coin landing on its edge, to demonstrate that the ...probability of heads or tails is less than 50%, depending on the initial toss conditions, the coin geometry and conditions of the coin and landing surfaces. For the dynamic model with rebounds that is the subject of this second part of the paper, it is found that the probability that a 50 Euro cent coin thrown from a normal height with common initial velocity conditions and appropriate surface conditions will end up on its edge is in the order of one against several thousand.
Considering that a fair coin has two sides and a cylindrical edge, the probability that it would fall on its edge is calculated, yielding the probability of heads or tails of less than 50%. In this ...first part, the theoretical models for a static case and for five dynamic cases, without rebounds, show that there is a small probability that the coin does not fall on its head or tail, depending on the initial toss conditions, the coin geometry and conditions of the coin and landing surfaces. It is found that the probability that a 50 Eurocent coin thrown from a normal height with common initial velocity conditions and appropriate surface conditions will end up on its edge is in the order of one against several thousand. In the second part of the paper, the dynamic model with rebounds is investigated.
Long bone fractures in hostile environments pose unique challenges due to limited resources, restricted access to healthcare facilities, and absence of surgical expertise. While external fixation has ...shown promise, the availability of trained surgeons is limited, and the procedure may frighten unexperienced personnel. Therefore, an easy-to-use external fixator (EZExFix) that can be performed by nonsurgeon individuals could provide timely and life-saving treatment in hostile environments; however, its efficacy and accuracy remain to be demonstrated. This study tested the learning curve and surgical performance of nonsurgeon analog astronauts (
= 6) in managing tibial shaft fractures by the EZExFix during a simulated Mars inhabited mission, at the Mars Desert Research Station (Hanksville, UT, USA). The reduction was achievable in the different 3D axis, although rotational reductions were more challenging. Astronauts reached similar bone-to-bone contact compared to the surgical control, indicating potential for successful fracture healing. The learning curve was not significant within the limited timeframe of the study (N = 4 surgeries lasting <1 h), but the performance was similar to surgical control. The results of this study could have important implications for fracture treatment in challenging or hostile conditions on Earth, such as war or natural disaster zones, developing countries, or settings with limited resources.
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