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Mersenne prime

In mathematics, a Mersenne prime is a prime number, one less than a power of two. That is, it is a prime number of the form Mn = 2n − 1 for some integer n, they are named after Marin Mersenne, a French Minim friar, who studied them in the early 17th century. The exponents n which give Mersenne primes are 2, 3, 5, 7, 13, 17, 19, 31... and the resulting Mersenne primes are 3, 7, 31, 127, 8191, 131071, 524287, 2147483647.... If n is a composite number so is 2n − 1; this definition is therefore equivalent to the definition as a prime number of the form Mp = 2p − 1 for some prime p. More numbers of the form Mn = 2n − 1 without the primality requirement may be called Mersenne numbers. Sometimes, Mersenne numbers are defined to have the additional requirement that n be prime; the smallest composite Mersenne number with prime exponent n is 211 − 1 = 2047 = 23 × 89. Mersenne primes Mp are noteworthy due to their connection to perfect numbers; as of February 2020, 51 Mersenne primes are known. The largest known prime number, 282,589,933 − 1, is a Mersenne prime.

Since 1997, all newly found Mersenne primes have been discovered by the Great Internet Mersenne Prime Search, a distributed computing project on the Internet. Many fundamental questions about Mersenne primes remain unresolved, it is not known whether the set of Mersenne primes is finite or infinite. The Lenstra–Pomerance–Wagstaff conjecture asserts that there are infinitely many Mersenne primes and predicts their order of growth, it is not known whether infinitely many Mersenne numbers with prime exponents are composite, although this would follow from believed conjectures about prime numbers, for example, the infinitude of Sophie Germain primes congruent to 3. For these primes p, 2p + 1 will divide Mp, for example, 23 | M11, 47 | M23, 167 | M83, 263 | M131, 359 | M179, 383 | M191, 479 | M239, 503 | M251. Since for these primes p, 2p + 1 is congruent to 7 mod 8, so 2 is a quadratic residue mod 2p + 1, the multiplicative order of 2 mod 2p + 1 must divide − 1 2 = p. Since p is a prime, it must be p or 1.

However, it cannot be 1 since Φ 1 = 1 and 1 has no prime factors, so it must be p. Hence, 2p + 1 divides Φ p = 2 p − 1 and 2 p − 1 = M p cannot be prime; the first four Mersenne primes are M2 = 3, M3 = 7, M5 = 31 and M7 = 127 and because the first Mersenne prime starts at M2, all Mersenne primes are congruent to 3. Other than M0 = 0 and M1 = 1, all other Mersenne numbers are congruent to 3. In the prime factorization of a Mersenne number there must be at least one prime factor congruent to 3. A basic theorem about Mersenne numbers states that if Mp is prime the exponent p must be prime; this follows from the identity 2 a b − 1 = ⋅ = ⋅. This rules out primality for Mersenne numbers with composite exponent, such as M4 = 24 − 1 = 15 = 3 × 5 = ×. Though the above examples might suggest that Mp is prime for all primes p, this is not the case, the smallest counterexample is the Mersenne number M11 = 211 − 1 = 2047 = 23 × 89; the evidence at hand suggests that a randomly selected Mersenne number is much more to be prime than an arbitrary randomly selected odd integer of similar size.

Nonetheless, prime values of Mp appear to grow sparse as p increases. For example, eight of the first 11 primes p give rise to a Mersenne prime Mp (the correct terms on Mer

Crescent-chested puffbird

The crescent-chested puffbird is a species of bird in the family Bucconidae, the puffbirds. It is endemic to Brazil, its natural habitats are subtropical and tropical moist lowland forests and subtropical and tropical moist montane forests. This elegant colored bird, has a chest band composed of a bright-white throat, a narrow band of black, a light cinnamon band blending into the chest, its main range is southeast Brazil, including the Atlantic coast, it inhabits a small disjunct range in northern and coastal Maranhão state, Northeast Region, Brazil. Crescent-chested puffbird videos in the Internet Bird Collection Crescent-chested puffbird photo gallery VIREO Photo-High Res

Moral treatment

Moral treatment was an approach to mental disorder based on humane psychosocial care or moral discipline that emerged in the 18th century and came to the fore for much of the 19th century, deriving from psychiatry or psychology and from religious or moral concerns. The movement is associated with reform and development of the asylum system in Western Europe at that time, it fell into decline as a distinct method by the 20th century, due to overcrowding and misuse of asylums and the predominance of biomedical methods. The movement is seen as influencing certain areas of psychiatric practice up to the present day; the approach has been praised for freeing sufferers from shackles and barbaric physical treatments, instead considering such things as emotions and social interactions, but has been criticised for blaming or oppressing individuals according to the standards of a particular social class or religion. Moral treatment developed in the context of the Enlightenment and its focus on social welfare and individual rights.

At the start of the 18th century, the "insane" were viewed as wild animals who had lost their reason. They were not held morally responsible but were subject to scorn and ridicule by the public, sometimes kept in madhouses in appalling conditions in chains and neglected for years or subject to numerous tortuous "treatments" including whipping, bloodletting, starvation, irritant chemicals, isolation. There were some attempts to argue for more psychological understanding and therapeutic environments. For example, in England John Locke popularized the idea that there is a degree of madness in most people because emotions can cause people to incorrectly associate ideas and perceptions, William Battie suggested a more psychological approach, but conditions remained poor; the treatment of King George III led to increased optimism about the possibility of therapeutic interventions. Under the Enlightened concern of Grand Duke Pietro Leopoldo in Florence, Italian physician Vincenzo Chiarugi instituted humanitarian reforms.

Between 1785 and 1788 he managed to outlaw chains as a means of restraint at the Santa Dorotea hospital, building on prior attempts made there since the 1750s. From 1788 at the newly renovated St. Bonifacio Hospital he did the same, led the development of new rules establishing a more humane regime; the ex-patient Jean-Baptiste Pussin and his wife Margueritte, the physician Philippe Pinel, are recognized as the first instigators of more humane conditions in asylums. From the early 1780s, Pussin had been in charge of the mental hospital division of the La Bicêtre, an asylum in Paris for male patients. From the mid-1780s, Pinel was publishing articles on links between emotions, social conditions and insanity. In 1792, Pinel became the chief physician at the Bicetre. Pussin showed Pinel how knowing the patients meant they could be managed with sympathy and kindness as well as authority and control. In 1797, Pussin first freed patients of their chains and banned physical punishment, although straitjackets could be used instead.

Patients were allowed to move about the hospital grounds, dark dungeons were replaced with sunny, well-ventilated rooms. Pussin and Pinel's approach was seen as remarkably successful and they brought similar reforms to a mental hospital in Paris for female patients, La Salpetrière. Pinel's student and successor, Jean Esquirol, went on to help establish 10 new mental hospitals that operated on the same principles. There was an emphasis on the selection and supervision of attendants in order to establish a suitable setting to facilitate psychological work, on the employment of ex-patients as they were thought most to refrain from inhumane treatment while being able to stand up to pleading, menaces, or complaining. Pinel used the term "traitement moral" for the new approach. At that time "moral", in French and internationally, had a mixed meaning of either psychological/emotional or moral. Pinel distanced himself from the more religious work, developed by the Tukes, in fact considered that excessive religiosity could be harmful.

He sometimes took a moral stance himself, however, as to what he considered to be mentally healthy and appropriate. An English Quaker named William Tuke independently led the development of a radical new type of institution in northern England, following the death of a fellow Quaker in a local asylum in 1790. In 1796, with the help of fellow Quakers and others, he founded the York Retreat, where about 30 patients lived as part of a small community in a quiet country house and engaged in a combination of rest and manual work. Rejecting medical theories and techniques, the efforts of the York Retreat centered around minimizing restraints and cultivating rationality and moral strength; the entire Tuke family became known as founders of moral treatment. They created a family-style ethos and patients performed chores to give them a sense of contribution. There was a daily routine of both leisure time. If patients behaved well, they were rewarded; the patients were told. In this sense, the patient's moral autonomy was recognized.

William Tuke's grandson, Samuel Tuke, published an influential work in the early 19th century on the methods of the retreat. A different background to the moral approach may be discerned in Scotland. Interest in mental illness was a feature of the Edinb