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1728 has twenty-eight divisors, which is a perfect count (as with 12, with six divisors). It also has a Euler totient of 576 or 24 2, which divides 1728 thrice over. [10] 1728 is an abundant and semiperfect number, as it is smaller than the sum of its proper divisors yet equal to the sum of a subset of its proper divisors. [11] [12]
Euler was aware of the equality 59 4 + 158 4 = 133 4 + 134 4 involving sums of four fourth powers; this, however, is not a counterexample because no term is isolated on one side of the equation. He also provided a complete solution to the four cubes problem as in Plato's number 3 3 + 4 3 + 5 3 = 6 3 or the taxicab number 1729.
1728 was a leap year starting on Thursday of the Gregorian calendar and a leap year starting on Monday of the Julian calendar, the 1728th year of the Common Era (CE) and Anno Domini (AD) designations, the 728th year of the 2nd millennium, the 28th year of the 18th century, and the 9th year of the 1720s decade. As of the start of 1728, the ...
1729 is the natural number following 1728 and preceding 1730. It is the first nontrivial taxicab number , expressed as the sum of two cubic numbers in two different ways. It is known as the Ramanujan number or Hardy–Ramanujan number after G. H. Hardy and Srinivasa Ramanujan .
His formula is again mentioned by Eutokios in a commentary on Archimedes. [4] In 499 CE Aryabhata, a mathematician-astronomer from the classical age of Indian mathematics and Indian astronomy, gave a method for finding the cube root of numbers having many digits in the Aryabhatiya (section 2.5). [5]
An alternative and equivalent definition of Carmichael numbers is given by Korselt's criterion.. Theorem (A. Korselt 1899): A positive composite integer is a Carmichael number if and only if is square-free, and for all prime divisors of , it is true that .
Klein's j-invariant in the complex plane. In mathematics, Felix Klein's j-invariant or j function, regarded as a function of a complex variable τ, is a modular function of weight zero for special linear group SL(2, Z) defined on the upper half-plane of complex numbers.
[3] 31 May – The Royal Bank of Scotland extends the first overdraft (to Edinburgh merchant William Hogg for £1,000). [4] 14 June – Congress of Soissons begins in an effort to end the Anglo-Spanish War. Late Summer – Voltaire ends his exile in England.