Proof that $1729$ is the smallest taxicab number For homework I have to produce the proof (algebraic or otherwise) to show that $1729$ HAS to be the smallest taxi cab number A taxicab number means that it is the sum of two different cubes and can be made with $2$ sets of numbers
Generalised Hardy-Ramanujan Numbers - Mathematics Stack Exchange The number 1729 is famously the smallest positive integer expressible as the sum of two positive cubes in two different ways ($1729=1^3+12^3=9^3+10^3$) There is plenty of work on "taxicab numbers"
1729, and related questions - Mathematics Stack Exchange I had ridden in taxi cab number 1729 and remarked that the number seemed to me rather a dull one, and that I hoped it was not an unfavorable omen ‘No,’ he replied, ‘it is a very interesting number; it is the smallest number expressible as the sum of two cubes in two different ways ’
Numbers that can be expressed as the sum of two cubes in exactly two . . . Since you have $$ 1729 = 10^3 + 9^3 = 12^3 + 1^3 $$ Multiply both sides by $ n^3 $ where $ n $ is a positive integer to get $$ 1729 n^3 = (10n)^3 + (9n)^3 = (12n)^3 + n^3 $$ and there you have it, plug in any value for $ n $ and and you have your proof for the infinitude of such numbers Obviously this doesn't include all the solutions but it
A number is a pseudoprime - Mathematics Stack Exchange $2^{1729}-2=2(2^{1728}-1) $ And $1729=7\cdot 13 \cdot 19 $ Check that each of $6,12,18$ (each one less than the primes $7,13,19$) divide the exponent $1728 $ So (using Fermat's "Little Theorem" here) $2^{1728}-1$ is divisible by the product $7 \cdot 13 \cdot 19 = 1729 $ Hence if we double this we get $1729$ being a divisor of $2^{1729}-2 $
For any integer $a$, $a^{37} \\equiv a \\left( \\text{mod } 1729\\right)$ We compute $\lambda(1729)= lcm[\lambda(7),\lambda(13),\lambda(19)]=lcm(6,12,18)=36$ and the result follows As an added bonus, you cannot get any better than this $\lambda(n)$ is the lowest number such that $\forall (a,n)=1, \ a^{\lambda(n)} \equiv 1 \mod n$
A curious case of $1729$ - Mathematics Stack Exchange Ramanujan's Taxicab number 1729 is famous for being the smallest positive integer which can be written as the sum of two positive cubes in two different ways On a different note, I observed that $12^3 + 1^3 = 1729$ $12^2 + 1^2 = 1\cdot7\cdot2\cdot9 + 1 + 7 + 2 + 9$ $12^1 + 1^1 = -1+7-2+9$
If $\\gcd(a,1729) \\neq 1$, show that $a^{37 }\\equiv a \\pmod {1729}$ Stack Exchange Network Stack Exchange network consists of 183 Q A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers