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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.
The square root of a positive integer is the product of the roots of its prime factors, because the square root of a product is the product of the square roots of the factors. Since p 2 k = p k , {\textstyle {\sqrt {p^{2k}}}=p^{k},} only roots of those primes having an odd power in the factorization are necessary.
11 2 = 121 12 2 = 144 13 2 ... For example, the square of 55376 is 3066501376, both ... Methods of computing square roots – Algorithms for calculating square roots;
A square has even multiplicity for all prime factors (it is of the form a 2 for some a). The first: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 (sequence A000290 in the OEIS ). A cube has all multiplicities divisible by 3 (it is of the form a 3 for some a ).
The square root of 2 (approximately 1.4142) ... By steps 5 and 8, a and b are both even, which contradicts step 3 (that is irreducible). Since we have derived a ...
All square triangular numbers have the form , where is a convergent to the continued fraction expansion of , the square root of 2. [ 4 ] A. V. Sylwester gave a short proof that there are infinitely many square triangular numbers: If the n {\displaystyle n} th triangular number n ( n + 1 ) 2 {\displaystyle {\tfrac {n(n+1)}{2}}} is square, then ...
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
a square (11 times 11) the sum of the powers of 3 from 0 to 4, so a repunit in ternary. Furthermore, 121 is the only square of the form + + + +, where p is prime (3, in this case). [1] the sum of three consecutive prime numbers (37 + 41 + 43).