| ||||
|---|---|---|---|---|
| Cardinal | one hundred forty-four | |||
| Ordinal | 144th (one hundred forty-fourth) | |||
| Factorization | 24 × 32 | |||
| Divisors | 1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 36, 48, 72, 144 | |||
| Greek numeral | ΡΜΔ´ | |||
| Roman numeral | CXLIV | |||
| Binary | 100100002 | |||
| Ternary | 121003 | |||
| Senary | 4006 | |||
| Octal | 2208 | |||
| Duodecimal | 10012 | |||
| Hexadecimal | 9016 | |||
144 (one hundred [and] forty-four) is the natural number following 143 and preceding 145. It is coincidentally both the square of twelve (a dozen dozens, or one gross.) and the twelfth Fibonacci number, and the only nontrivial number in the sequence that is square. [1] [2]
144 is a highly totient number. [3]
144 is the smallest number whose fifth power is a sum of four (smaller) fifth powers. This solution was found in 1966 by L. J. Lander and T. R. Parkin, and disproved Euler's sum of powers conjecture. It was famously published in a paper by both authors, whose body consisted of only two sentences: [4]
A direct search on the CDC 6600 yielded
275 + 845 + 105 + 1335 = 1445
as the smallest instance in which four fifth powers sum to a fifth power. This is a counterexample to a conjecture by Euler that at least nnth powers are required to sum to an nth power, n > 2.
In number theory, Euler's conjecture is a disproved conjecture related to Fermat's Last Theorem. It was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n many kth powers of positive integers is itself a kth power, then n is greater than or equal to k:
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy."
111 is the natural number following 110 and preceding 112.
72 (seventy-two) is the natural number following 71 and preceding 73. It is half a gross or six dozen.
34 (thirty-four) is the natural number following 33 and preceding 35.
58 (fifty-eight) is the natural number following 57 and preceding 59.
64 (sixty-four) is the natural number following 63 and preceding 65.
120 is the natural number following 119 and preceding 121. It is five sixths of a gross, or ten dozens.
1000 or one thousand is the natural number following 999 and preceding 1001. In most English-speaking countries, it can be written with or without a comma or sometimes a period separating the thousands digit: 1,000.
360 is the natural number following 359 and preceding 361.
2000 is a natural number following 1999 and preceding 2001.
100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.
100,000,000 is the natural number following 99,999,999 and preceding 100,000,001.
168 is the natural number following 167 and preceding 169.
In mathematics and statistics, sums of powers occur in a number of contexts:
In number theory, the Lander, Parkin, and Selfridge conjecture concerns the integer solutions of equations which contain sums of like powers. The equations are generalisations of those considered in Fermat's Last Theorem. The conjecture is that if the sum of some k-th powers equals the sum of some other k-th powers, then the total number of terms in both sums combined must be at least k.
In arithmetic and algebra, the fifth power or sursolid of a number n is the result of multiplying five instances of n together:
In arithmetic and algebra the sixth power of a number n is the result of multiplying six instances of n together. So:
