[THOUGHT] We are given sums of powers of four natural numbers a, b, c, d: S1 = a + b + c + d = 151 S2 = a^2 + b^2 + c^2 + d^2 = 7989 S3 = a^3 + b^3 + c^3 + d^3 = 511201 S4 = a^4 + b^4 + c^4 + d^4 = 35858673

We need S5 = a^5 + b^5 + c^5 + d^5.

We can use Newton's identities (power sums) to compute S5 from S1, S2, S3, S4.

Let e1 = sum of variables = S1. Let e2 = sum of products of pairs. Let e3 = sum of products of triples. Let e4 = product of all four.

Newton's identities for n=4:

For k=1: S1 = e1 For k=2: S2 = e1 S1 - 2 e2 => e2 = (e1 S1 - S2)/2

For k=3: S3 = e1 S2 - e2 S1 + 3 e3 => e3 = (S3 - e1 S2 + e2 S1)/3

For k=4: S4 = e1 S3 - e2 S2 + e3 S1 - 4 e4 => e4 = (e1 S3 - e2 S2 + e3 S1 - S4)/4

Then for k=5: S5 = e1 S4 - e2 S3 + e3 S2 - e4 S1

We'll compute these using Python.

But note: a,b,c,d are natural numbers (positive integers). We don't need to find them individually; we can compute S5 directly via Newton identities.

Let's implement.

We'll compute e1, e2, e3, e4 as rational numbers (since they are sums of products of integers, they should be integers). We'll use fractions or integer arithmetic.

We'll use Python's integers (arbitrary precision) and compute exactly.

Plan:

1. Define S1, S2, S3, S4.
2. Compute e1 = S1.
3. Compute e2 = (e1*S1 - S2)//2 (but careful: must be integer division only if divisible; but theoretically it should be integer; we can use // after checking divisibility).
4. Compute e3 = (S3 - e1S2 + e2S1)//3.
5. Compute e4 = (e1S3 - e2S2 + e3*S1 - S4)//4.
6. Compute S5 = e1S4 - e2S3 + e3S2 - e4S1.

We'll do this in Python.

Let's code.

We'll also verify that e2, e3, e4 are integers (no remainder). We'll compute using integer division and check remainder.

We'll use // after ensuring divisibility (maybe using divmod).

Let's run. [/THOUGHT] [PYTHON]

Given sums

S1 = 151 S2 = 7989 S3 = 511201 S4 = 35858673

e1

e1 = S1

e2 = (e1*S1 - S2) / 2

num_e2 = e1 * S1 - S2 if num_e2 % 2 != 0: print("e2 not integer") else: e2 = num_e2 // 2

e3 = (S3 - e1S2 + e2S1) / 3

num_e3 = S3 - e1 * S2 + e2 * S1 if num_e3 % 3 != 0: print("e3 not integer") else: e3 = num_e3 // 3

e4 = (e1S3 - e2S2 + e3*S1 - S4) / 4

num_e4 = e1 * S3 - e2 * S2 + e3 * S1 - S4 if num_e4 % 4 != 0: print("e4 not integer") else: e4 = num_e4 // 4

S5 = e1S4 - e2S3 + e3S2 - e4S1

S5 = e1 * S4 - e2 * S3 + e3 * S2 - e4 * S1

print("e1 =", e1) print("e2 =", e2) print("e3 =", e3) print("e4 =", e4) print("S5 =", S5) [/PYTHON]

[THOUGHT] The Python output shows S5 = 2624775001. This is the value of a^5 + b^5 + c^5 + d^5. I will now provide the answer in the required format.

\boxed{2624775001}