︠4206ffba-6eb6-41fb-b22c-95aaf4ae8e5ai︠
%md
**Question:** I am looking for two integers, $a$ and $b$, such that $a^4+ b^4 = 11572060353961555386606814001$. Can anyone help me to solve this problem using SageMath .
︡0ec26d29-209f-4c5d-b561-d6e15767013b︡{"done":true,"md":"**Question:** I am looking for two integers, $a$ and $b$, such that $a^4+ b^4 = 11572060353961555386606814001$. Can anyone help me to solve this problem using SageMath ."}
︠3249eb1f-fa34-4480-9f5a-235604c46fbes︠
n = 11572060353961555386606814001
factor(n)
︡67fe4211-2850-4d40-aec4-1b1e5b7fac48︡{"stdout":"101 * 44621 * 10147901 * 253031037087781\n"}︡{"done":true}
︠fc01dc90-840e-40c1-a72c-14b690aec8e1︠
# n is a sum of two *squares*:
sum_of_k_squares(2, n)
︡a0efb553-f130-408b-8add-60d946fec40f︡{"stdout":"(70463059530200, 81283562887001)\n"}︡{"done":true}
︠9e108978-d42c-4eab-a267-fa19d85417ac︠
#but not a perfect square
N(sqrt(70463059530200))
︡e902006c-a89c-47d4-b38b-43246582ea95︡{"stdout":"8.39422775067486e6\n"}︡{"done":true}
︠fbcab688-09fe-40aa-855d-ddc6708a4af7s︠
for p, _ in factor(n):
    print p, p%4
︡21263098-0257-438e-a962-1c11af946e62︡{"stdout":"101 1\n44621 1\n10147901 1\n253031037087781 1\n"}︡{"done":true}
︠12d3b20f-52b1-4ac6-ad9d-af979718ccd2i︠
%md
Because each $p$ dividing $n$ exactly divides $n$ and is $1$ mod $4$, there are 16=$2^4$ ways to write $n$ as a sum of two **squares**.
If you write $n$ as a sum of two fourth powers, that's just one of these: $$(a^2)^2 + (b^2)^2 = n$$
︡535a50cc-c631-49f4-a6df-9020be2e99bc︡{"done":true,"md":"Because each $p$ dividing $n$ exactly divides $n$ and is $1$ mod $4$, there are 16=$2^4$ ways to write $n$ as a sum of two **squares**.\nIf you write $n$ as a sum of two fourth powers, that's just one of these: $$(a^2)^2 + (b^2)^2 = n$$"}
︠0b3debfa-9a83-4611-8f40-71eb89a6ffe4s︠
K.<i> = QuadraticField(-1)
v = [factor(K(p)) for p, _ in factor(n)]
v
︡b1e42bb6-44dd-4aae-a4ac-03e0b6ef02bd︡{"stdout":"[(-1) * (i - 10) * (i + 10), (-10*i + 211) * (10*i + 211), (-1099*i + 2990) * (1099*i + 2990), (-5563791*i + 14902190) * (5563791*i + 14902190)]\n"}︡{"done":true}
︠6a98600c-adca-41d9-b757-b858d8d1bcc1s︠
v[0][0][0]
︡3d6924f7-0ff0-4e55-b132-9d759844e89b︡{"stdout":"i - 10\n"}︡{"done":true}
︠3f38947b-fa25-43a2-861f-83bad1412f23s︠
for c in cartesian_product_iterator([[0,1]]*4):
    rep = prod(v[i][k][0] for i, k in enumerate(c))
    print c, rep
    if is_square(rep[0]):
        print "FOUND ONE! ", rep
︡105d078d-767a-4fab-b3a8-9820f5bcc460︡{"stdout":"(0, 0, 0, 0) 81283562887001*i - 70463059530200\n(0, 0, 0, 1) 15216760633199*i - 106491833253980\n(0, 0, 1, 0) 16301319409199*i - 106331215263820\n(0, 0, 1, 1) -57372053092999*i - 90997295992000\n(0, 1, 0, 0) 74255239366801*i - 77834566746020\n(0, 1, 0, 1) 5077165185799*i - 107453630686160\n(0, 1, 1, 0) 6172053092999*i - 107396304008000\n(0, 1, 1, 1) -65720906289199*i - 85163506447820\n(1, 0, 0, 0) -65720906289199*i + 85163506447820\n(1, 0, 0, 1) 6172053092999*i + 107396304008000\n(1, 0, 1, 0) 5077165185799*i + 107453630686160\n(1, 0, 1, 1) 74255239366801*i + 77834566746020\n(1, 1, 0, 0) -57372053092999*i + 90997295992000\n(1, 1, 0, 1) 16301319409199*i + 106331215263820\n(1, 1, 1, 0) 15216760633199*i + 106491833253980\n(1, 1, 1, 1) 81283562887001*i + 70463059530200\n"}︡{"done":true}
︠9f7b8562-ab85-4930-a125-e330a72ce1b9s︠
# e.g.,
5077165185799^2 + 107453630686160^2
︡f068da9f-cd06-4646-b7f1-5144d3625e75︡{"stdout":"11572060353961555386606814001\n"}︡{"done":true}
︠6bec8595-1bd0-4d65-bd02-c6acbfe5a80ci︠
%md
As you can see, we didn't find any with the rep as a sum of two squares with each a square itself.
So there are no such $a,b$.
︡e4dd1322-7338-4f8b-9f82-b409be6e4500︡{"done":true,"md":"As you can see, we didn't find any with the rep as a sum of two squares with each a square itself.\nSo there are no such $a,b$."}
︠34ac3077-a186-48e1-92fb-4a0307129ca3si︠
%md Same problem by brute force search...
︡7acde03b-dafa-478b-9410-b0668c672645︡{"md":"Same problem by brute force search..."}︡{"done":true}
︠ac135b0a-22ce-4596-b425-6be2eb2e14e7︠
%python
from math import sqrt
def sum_of_two_fourth_powers(m):
    b = int(sqrt(sqrt(m)))
    print(b)
    for i in xrange(b):
        s = sqrt(sqrt(m-i**4))
        if int(s) == s:
            return i, int(s)
︡d8fabed9-6a2b-475e-b37f-96e0632338f8︡{"done":true}
︠d3c957f7-e7cb-498e-869c-7faeb1ce5328︠
# test the above function
%time sum_of_two_fourth_powers(848^4 + 798^4)
︡1a86fe3c-d337-4815-a0b4-d7e178450975︡{"stdout":"980\n(798, 848)\n"}︡{"stdout":"\nCPU time: 0.00 s, Wall time: 0.00 s\n"}︡{"done":true}
︠fca3f099-ad74-4279-95ea-3ac4009bb105︠
# now try it on yours... and get nothing.
%time sum_of_two_fourth_powers(11572060353961555386606814001)
︡7b24f9a6-72a5-4267-827a-e0fb20fdd70f︡{"stdout":"10371765\n"}︡{"stdout":"\nCPU time: 23.28 s, Wall time: 23.68 s"}︡{"stdout":"\n"}︡{"done":true}
︠b368cdec-b04f-4846-b77a-df3f7be6ce38︠











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