## Integers N such that N = a^2 + b^2 = c^3 + d^3

$N \; = \; a^2 \; + \; b^2 \; = \; c^3 \; + \; d^3$

Here are the first six integers   $> 2$,   which are sums of two cubes of positive integers

$9 \; = \; 1^3 \; + \; 2^3$
$16 \; = \; 2^3 \; + \; 2^3$
$28 \; = \; 1^3 \; + \; 3^3$
$35 \; = \; 2^3 \; + \; 3^3$
$54 \; = \; 3^3 \; + \; 3^3$
$65 \; = \; 1^3 \; + \; 4^3$

None of the numbers   9, 16, 28, 35,   and   54   is a sum of two squares of integers,

while     $65 \; = \; 1^2 \; + \; 8^2$

Hence, the least integer   $> 2$   which is a sum of two squares of integers and a sum
of two cubes of positive integers is   65.

Find few other positive integers which are sums of two squares and sums of two cubes of two relatively prime positive integers.

math grad - Interest: Number theory
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### 5 Responses to Integers N such that N = a^2 + b^2 = c^3 + d^3

1. K.D. BAJPAI says:

Few terms are :
72 = 2^3 + 4^3 = 6^2 + 6^2
370 = 3^3 + 4^3 = 3^2 + 19^2 = 9^2 + 17^2
468 = 5^3 + 7^3 = 12^2 + 18^2

2. K.D. BAJPAI says:

Also,
128 = 4^3 + 4^3 = 8^2 + 8^2
250 = 5^3 + 5^3 = 5^2 + 15^2 = 9^2 +13^2

3. K.D. BAJPAI says:

I am not sure, such terms are acceptable:
1024 = 8^3 + 8^3 = 0^2 + 32^2

If not acceptable, it needs to be specified in the definition itself.

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• benvitalis says:

I asked to find few other positive integers which are sums of two squares and sums of two cubes of two relatively prime positive integers.

4. K.D. BAJPAI says:

I could not consider the condition relatively prime in my above submission. As such, only one equation meets the required condition:
370 = 3^3 + 4^3 = 3^2 + 19^2