Euler's totient function

In number theory, Euler's totient function counts the positive integers up to a given integer n that are prime to n. It is written using the Greek letter phi as φ or ϕ, may be called Euler's phi function. In other words, it is the number of integers k in the range 1 ≤ k ≤ n for which the greatest common divisor gcd is equal to 1; the integers k of this form are sometimes referred to as totatives of n. For example, the totatives of n = 9 are the six numbers 1, 2, 4, 5, 7 and 8, they are all prime to 9, but the other three numbers in this range, 3, 6, 9 are not, since gcd = gcd = 3 and gcd = 9. Therefore, φ = 6; as another example, φ = 1 since for n = 1 the only integer in the range from 1 to n is 1 itself, gcd = 1. Euler's totient function is a multiplicative function, meaning that if two numbers m and n are prime φ = φφ; this function gives the order of the multiplicative group of integers modulo n. It is used for defining the RSA encryption system. Leonhard Euler introduced the function in 1763.

However, he did not at that time choose any specific symbol to denote it. In a 1784 publication, Euler studied the function further, choosing the Greek letter π to denote it: he wrote πD for "the multitude of numbers less than D, which have no common divisor with it"; this definition varies from the current definition for the totient function at D = 1 but is otherwise the same. The now-standard notation φ comes from Gauss's 1801 treatise Disquisitiones Arithmeticae, although Gauss didn't use parentheses around the argument and wrote φA. Thus, it is called Euler's phi function or the phi function. In 1879, J. J. Sylvester coined the term totient for this function, so it is referred to as Euler's totient function, the Euler totient, or Euler's totient. Jordan's totient is a generalization of Euler's; the cototient of n is defined as n − φ. It counts the number of positive integers less than or equal to n that have at least one prime factor in common with n. There are several formulas for computing φ.

It states φ = n ∏ p ∣ n. The proof of Euler's product formula depends on two important facts; this means that if gcd = 1 φ = φ φ. If p is prime and k ≥ 1 φ = p k − 1 = p k. Proof: since p is a prime number the only possible values of gcd are 1, p, p2... pk, the only way for gcd to not equal 1 is for m to be a multiple of p. The multiples of p that are less than or equal to pk are p, 2p, 3p... pk − 1p = pk, there are pk − 1 of them. Therefore, the other pk − pk − 1 numbers are all prime to pk; the fundamental theorem of arithmetic states that if n > 1 there is a unique expression for n, n = p 1 k 1 ⋯ p r k r, where p1 < p2 <... < pr are prime numbers and each ki ≥ 1. Using the multiplicative property of φ and the formula for φ gives φ = φ φ ⋯ φ = p 1 k 1 p 2 k 2 ⋯ p r k r = p 1 k 1 p 2 k 2 ⋯ p r k

Gordon Parker (psychiatrist)

Gordon Barraclough Parker AO is an Australian psychiatrist, scientia professor of psychiatry at the University of New South Wales. Parker’s particular focus is on the phenomenology and epidemiology of mood disorders, social psychiatry, the treatment and management of mood disorders, his research has assisted in modelling psychiatric conditions – depression and personality disorders – and examining causes and treatments for mood disorders, together with innovative clinical work. Parker is a critic of the current unitary classification of major depressive disorder, has proposed the revival of the diagnosis of melancholia. In 2010 he was made an officer of the Order of Australia in recognition of his distinguished service to psychiatry as a clinician and researcher as a major contributor to the understanding and innovative treatment of mood disorders and as founder and Executive Director of the Black Dog Institute. Parker was born in Melbourne, schooled at Shore in Sydney, completed an MB BS at the University of Sydney, an MD, PhD and DSc at the University of New South Wales.

He is married, with four children. His path to Medicine and his clinical rationale is outlined in his autobiography, "A Piece of My Mind". Parker was Head of the School of Psychiatry at the University of NSW from 1983-2002, as well as Director of the Division of Psychiatry at Prince of Wales and Prince Henry Hospitals in Sydney from 1983 to 1996. In 2002, he became the founder and inaugural Director of the Black Dog Institute, an organisation that focuses on research into and treatment of mood disorders, in particular clinical depression and bipolar disorder; as a consequence of his advocacy for diagnosing melancholia, in 2010 Parker was invited to head a group of prominent international psychiatrists to argue for its separate status in the new DSM-5 classificatory system. Parker has been a member of the Editorial Boards of 16 journals, was the invited Editor of the December 2015 issue of Acta Psychiatrica Scandinavica, he is an invited assessor for various National Health and Medical Research Grants as well as for scientific journals, for example, The Lancet and The American Journal of Psychiatry.

He has been involved with the Royal Australian and New Zealand College of Psychiatrists – as Editor of the Journal and Chair of the Quality Assurance Committee. Parker has held a number of positions with legal organisations, including the NSW Guardianship Board and the NSW Administrative Appeals Tribunal, he has sought to make the community more aware of depressive sub-types and the bipolar conditions via multiple TV, radio and print interviews and personalised programs. During his time at the Black Dog Institute, Parker ‘translated’ research findings via educational programs – some with health professionals in mind and some shaped for the general community. Parker has developed or evaluated a number of assessment and self-assessment tools to help practitioners and individuals to gauge the type and clinical import of a mood disorder. In 2004, Parker received a Citation Laureate from the Institute for Scientific Information as the Australian Scientist most cited in the field of “Psychiatry/Psychology”.

His citations exceed 35,000. In 2008, Parker received a Human Rights and Equal Opportunity Commission award for his book: “Journeys with the Black Dog”, in 2010 his book “Tackling Depression at Work” was short-listed for a further Australian Human Rights Commission award. In 2017, Parker was awarded the James Cook medal of The Royal Society of New South Wales for "Outstanding contributions to both science and human welfare in and for the Southern Hemisphere". Lifetime awards include the RANZCP Senior Psychiatric Research Award, Australasian Society of Psychiatric Research Founders Award, NSW Forensic Psychology Award, UNSW Alumni Award for Science and Technology, Suicide Prevention Lifetime Research Award, Officer of the Order of Australia in 2010; the University of NSW awards Scientia Professor status for those academics who have demonstrated "outstanding research performance." Parker was the first clinical academic to be awarded such status in 2000, for a five-year period, subsequently for three further periods up until 2018.

Parker was a New South Wales finalist in the 2019 Australian of the Year Awards, in the category of Senior Australian of the Year. Parker has published more than 900 papers, over 600 in peer-reviewed journals. A complete list is available at the UNSW Faculty of Medicine website, Professor Gordon Barraclough Parker publications: Parker has worked as a creative writer – writing for The Mavis Bramston Show, a cartoonist for Oz magazine and The Bulletin, a book reviewer for major Australian newspapers, he was an ABC Science broadcaster in London. His play, Personality Games, was staged in 2004 at The Wharf Theatre in Sydney and La Mama's Carlton Courthouse in Melbourne. Bed and Bored, Lansdowne Press, Melbourne, 1966; the Bonds of Depression and Robertson, Sydney, 1978. Parental Overprotection and Stratton, NY, 1983; some Rules for Killing People, Ellard J Angus and Robertson, 1989. Melancholia: A Disorder of Movement and Mood, Cambridge University Press, NY, 1996. Dealing with Depression: a common sense guide to mood disorders, Allen & Unwin, Sydney, 2002.

Modelling and Managing the Depressive Disorders and Manicavasagar, Cambridge University Press, Cambridge


Darmai is an administrative unit, known as Union council, of Swat District in the Khyber Pakhtunkhwa province of Pakistan. District Swat has 9 Tehsils i.e. Khwazakhela, Madyan, Barikot and Kalam; each Tehsill comprises certain numbers of union councils. There are 65 union councils in 56 rural and 09 urban. Swat DistrictDarmai is an administrative unit, known as Union council, of Swat District in the Khyber Pakhtunkhwa province of Pakistan. District Swat has 9 Tehsils i.e. Khwazakhela, Madyan, Barikot and Kalam; each Tehsill comprises certain numbers of union councils. There are 65 union councils in 56 rural and 09 urban. Darmai Khyber-Pakhtunkhwa Government website section on Lower Dir United Nations Uploads PBS