Integer-Sided Triangles with integral medians

 
 

According to http://arxiv.org/vc/arxiv/papers/0901/0901.1857v1.pdf

… in Proposition 2, we prove that each integer-sided triangle can have at most two medians of integer length

 
Can you disprove this claim?

That is, find an integer-sided triangle with all three median lengths being integral.

 
 

Formulas for median length
 

MEDIANS 1

 
 

Paul found:

MEDIANS 2

 
 
 
 
 
 
 
 
 
 
 
 

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Primorial puzzle

 
 
                                                                   Part 1

 
 

To find   (m, n)   so that   C(m,n)   is a product of all prime numbers ≤ m

For example,

 
PRIMORIAL 1

 

It appears that the product of the primes is increasing faster than the possible combination numbers.

I don’t know if we could have another case where the combinations would equal the product of primes.

I don’t have the proof.

 
 
 
 
 

                                                                   Part 2

 
 

To find   (m, n)   so that   C(m,n)   is a product of consecutive prime numbers

 
Paul found:

 
PRIMORIAL 2

and

PRIMORIAL 3

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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when the sum of consecutive cubes is a square

 

Nicomachus’s Theorem
http://mathworld.wolfram.com/NicomachussTheorem.html

 
SUMS 2

 
 

To find a sum of four or more consecutive cubes – not starting with 1 – giving us the square of an integer

 
 

Paul found:

 
SUMS 1

 
 
From Republic of Math :

 
SUMS 3

 

From Derek :

 
SUMS 4

 
From Max Alekseyev:
A126200   Numbers n such that n^2 is a sum of consecutive cubes larger than 1
 
 
 
 
 
 
 
 
 
 
 
 
 

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Integers (A,B); A+B, A^2 + B^2 and A^3 + B^3 are all squares

 

RARE 1

 
 
David found:

 
RARE 2

 
 
 
 
 
 
 

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Equation | a^7 + b^3 = c^2, gcd(a,b,c) = 1

 
 
a7 + b3 = c2

 
David Radcliffe sent me a link to “primitive solutions to x^2 + y^3 = z^7″

http://arxiv.org/abs/math/0508174v1

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Difference of two consecutive cubes

 
 

Difference of 2 CUBES 1

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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Grid | Numbers from 1 to 25

 
 
Find the arrangement of the array :

 
5by5 multiplicative grid 1

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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