In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. Other positive integers are impolite. Polite numbers have also been called staircase numbers because the Young diagrams representing graphically the partitions of a polite number into consecutive integers (in the French style of drawing these diagrams) resemble staircases. If all numbers in the sum are strictly greater than one, the numbers so formed are also called trapezoidal numbers because they represent patterns of points arranged in a trapezoid.
The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied by Sylvester, Mason, and Leveque, as well as by many other more recent authors.
Read more about Polite Number: Examples and Characterization, Politeness, Construction of Polite Representations From Odd Divisors, Trapezoidal Numbers
Famous quotes containing the words polite and/or number:
“And anyone is free to condemn me to death
If he leaves it to nature to carry out the sentence.
I shall will to the common stock of air my breath
And pay a death tax of fairly polite repentance.”
—Robert Frost (1874–1963)
“Cultivated labor drives out brute labor. An infinite number of shrewd men, in infinite years, have arrived at certain best and shortest ways of doing, and this accumulated skill in arts, cultures, harvestings, curings, manufactures, navigations, exchanges, constitutes the worth of our world to-day.”
—Ralph Waldo Emerson (1803–1882)