Which numbers can be written as the sum of two cubes?

1729 is the smallest number which can be expressed as the sum of two cubes in two different ways: 1³ + 12³ and 9³ + 10³.

Can 2021 be written as the sum of two squares?

For if the number 4n + 1 were prime, it could certainly be written as the sum of two squares. Thus, since the numbers 21, 33, 57, 69, 77, 93, etc., which are contained in the form 4n + 1, are not sums of two squares, from this fact itself it is evident that they are not prime.

What is the smallest number that can be expressed as the sum of two different cubes in two different ways?

1729
It’s the smallest number expressible as the sum of two cubes in two different ways.” Because of this incident, 1729 is now known as the Ramanujan-Hardy number. To date, only six taxi-cab numbers have been discovered that share the properties of 1729.

How do you check whether a number can be represented as sum of cubes?

A number can be represented as the sum of the perfect cube of two consecutive numbers if the sum of the cube root of both consecutive numbers is equal to N.

What number Cannot be shown as pairs of cubes?

School cards; at least 5 minutes a day.

How do you determine whether a number can be written as a sum of two squares?

A number can be represented as a sum of two squares precisely when N is of the form n2∏pi where each pi is a prime congruent to 1 mod 4. If the equation a2+1≡a(modp) is solvable for some a, then p can be represented as a sum of two squares.

Why can’t you factor the sum of two squares?

It’s true that you can’t factor A²+B² on the reals — meaning, with real-number coefficients — if A and B are just simple variables. So it’s still true that a sum of squares can’t be factored as a sum of squares on the reals.

What is the sum of cubes?

A sum of cubes is a two-term expression where both terms are cubes and each term has the same sign. It is factored according to the following formula: a3 + b3 = (a + b) (a2 – ab + b2) How can you determine if an expression can be factored as a sum of cubes? 7:32.

What is the sum of cubes formula?

The sum of cubes (a3 + b3) formula is expressed as a3 + b3 = (a + b) (a2 – ab + b2).

What is Srinivasa Ramanujan number?

1729 is the natural number following 1728 and preceding 1730. It is a taxicab number, and is variously known as Ramanujan’s number and the Ramanujan-Hardy number, after an anecdote of the British mathematician G. H. Hardy when he visited Indian mathematician Srinivasa Ramanujan in hospital.

What is the sum of the two cubes 20683?

20683 = 10 3 + 27 3 = 19 3 + 24 3. In the paper Characterizing the Sum of Two Cubes, Kevin Broughan gives the relevant theorem, Theorem: Let N be a positive integer.

What is the formula to find the sum of two cubes?

The formula for the sum of two cubes is. a 3 + b 3 = ( a + b) ( a 2 − a b + b 2) a^3+b^3= (a+b) (a^2-ab+b^2) a ​ 3 ​ ​ + b ​ 3 ​ ​ = ( a + b) ( a ​ 2 ​ ​ − a b + b ​ 2 ​ ​) How to factor the sum of two cubes. YouTube. Take the course. Want to learn more about Algebra 2?

How do you find the difference between two cubes?

Rewrite the original problem as sum of two cubes, and then simplify. Since this is the “sum” case, the binomial factor and trinomial factor will have positive and negative middle signs, respectively. Example 2: Factor. y 3 − 8. {y^3} – 8 y3 − 8. This is a case of difference of two cubes since the number. 8.

What is the overall common factor of two cubes?

For the numbers, the greatest common factor is xy xy “. Therefore the overall common factor would be their product which is \\left ( 3 ight)\\left ( {xy} ight) = 3xy (3) (xy) = 3xy. After factoring it out, you’ll see that we have an easy problem on the difference of two cubes.