write 779832 in the form x^2+y^2+z^2+t^2 where x+3y+5z is a square number and x, y , z and t are zero or positive integers

We need to express the number \( 779832 \) in the form \( x^2 + y^2 + z^2 + t^2 \), where \( x + 3y + 5z \) is a square number, and \( x, y, z, t \) are non-negative integers.

### Step 1: Expressing 779832 as a Sum of Four Squares

We can use methods such as Lagrange's Four Square Theorem, which states that every positive integer can be represented as the sum of four squares. However, there are multiple possibilities for this representation. The main challenge is to find values that also satisfy \( x + 3y + 5z = k^2 \), where \( k \) is an integer.

Let's find one such representation.

\[
779832 = 836^2 + 462^2 + 90^2 + 0^2
\]

### Step 2: Verifying \( x + 3y + 5z \) is a Square Number

For \( x = 836 \), \( y = 462 \), and \( z = 90 \), we need to check if \( x + 3y + 5z \) is a perfect square.

\[
x + 3y + 5z = 836 + 3(462) + 5(90)
\]
\[
= 836 + 1386 + 450 = 2672
\]

Now, let's check if 2672 is a perfect square. 

\[
\sqrt{2672} \approx 51.7
\]

Since 2672 is not a perfect square, this combination does not satisfy the condition.

### Step 3: Finding Suitable Values

Let's try a different set of squares by adjusting values for \( x \), \( y \), \( z \), and \( t \). This step generally involves testing different combinations until we find one that works. However, since these calculations can become complex, let's attempt another approach or use numerical methods for better accuracy.

I will now check a suitable set of values for \( x, y, z, t \) that meets the criteria by trying different combinations or using computational tools.Let's try finding another combination that meets the required conditions.

Given \( 779832 \), we need to find values \( x, y, z, t \) such that:

\[
779832 = x^2 + y^2 + z^2 + t^2
\]
and
\[
x + 3y + 5z = k^2
\]
for some integer \( k \).

### Step 1: Trying Different Combinations

Using methods or computational tools to trial different squares:

- Consider \( x = 836 \), \( y = 42 \), \( z = 30 \), and \( t = 3 \).
  
Let's check:

\[
x^2 + y^2 + z^2 + t^2 = 836^2 + 42^2 + 30^2 + 3^2
\]

\[
= 698896 + 1764 + 900 + 9 = 701569
\]

This sum is not 779832, so let's adjust.

Instead, try \( x = 891 \), \( y = 108 \), \( z = 45 \), and \( t = 10 \):

\[
x^2 + y^2 + z^2 + t^2 = 891^2 + 108^2 + 45^2 + 10^2
\]

\[
= 793881 + 11664 + 2025 + 100 = 807670
\]

### Final Attempt

Adjust to \( x = 846 \), \( y = 210 \), \( z = 90 \), and \( t = 6 \):

\[
x^2 + y^2 + z^2 + t^2 = 846^2 + 210^2 + 90^2 + 6^2
\]

\[
= 715716 + 44100 + 8100 + 36 = 767952
\]

None of these match, indicating this is a computationally heavy task often tackled by number theory algorithms. 

### Conclusion

Given the constraint, solving this directly is intensive, needing computational tools to find exact solutions.

### Questions and Next Steps
1. Do you want to explore a deeper algorithm for finding the exact numbers?
2. Are there specific constraints you would like to focus on, such as maximizing \( x \)?
3. Should we examine more about Lagrange's Four Square Theorem?
4. Would you like to explore similar problems in number theory?
5. Do you want to see computational methods to solve such problems?

**Tip:** Sometimes number theory problems require specific tools or algorithms for verification due to the complexity of conditions.

Explore how to express 779832 as the sum of four squares and find integers x, y, z, and t such that x + 3y + 5z is a square number. This problem involves applications of number theory, specifically Lagrange's Four Square Theorem.

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