There are infinitely many sets of positive integers A<B<C<D such that A^2+B^2, A^2+B^2+C^2, and A^2+B^2+C^2+D^2 are all squares. Find the value of A≤ 1500 which leads to the largest number of solutions.
gap = 1000
mA = 1500
mB = mA + gap
mC = mA + 2*gap
mD = mA + 3*gap
largest_possible = int((mA**2 + mB**2 + mC**2 + mD**2)**.5)
squares = [n**2 for n in range(largest_possible+1)]
sumsquares = {}
adict_ab = {}
adict_abc = {}
adict_abcd = {}
counts_ab = [0 for i in range(mA+1)]
counts_abc = [0 for i in range(mA+1)]
counts_abcd = [0 for i in range(mA+1)]
def sumsq(*args):
if len(args) == 0:
return 0
elif len(args) == 1:
return squares[args[0]]
elif len(args) == 2:
ans = args[0]**2 + args[1]**2
else:
thekey = ''
for i,v in enumerate(args):
if i == 0:
thekey += str(v)
continue
if i == len(args) - 1:
break
thekey += ','+str(v)
ans = sumsquares[thekey] + squares[args[-1]]
return ans
for a in range(1, mA+1):
for b in range(a+1, mB+1):
mysum = sumsq(a,b)
if mysum in squares:
if a not in adict_ab:
adict_ab[a] = [[a,b]]
counts_ab[a] += 1
else:
adict_ab[a].append([a,b])
counts_ab[a] += 1
mykey = str(a)+','+str(b)
sumsquares[mykey] = mysum
maxcountab = max(counts_ab)
indices_b = [i for i, x in enumerate(counts_ab) if x == maxcountab]
print('ab', indices_b, counts_ab[indices_b[0]])
print(sum(counts_ab))
for key, vals in adict_ab.items():
a = key
for val in vals:
b = val[1]
for c in range(b + 1, mC+1):
mysum = sumsq(a,b,c)
if mysum in squares:
if a not in adict_abc:
adict_abc[a] = [[a,b,c]]
counts_abc[a] += 1
else:
adict_abc[a].append([a,b,c])
counts_abc[a] += 1
mykey = str(a)+','+str(b)+','+str(c)
sumsquares[mykey] = mysum
maxcountabc = max(counts_abc)
indices_c = [i for i, x in enumerate(counts_abc) if x == maxcountabc]
print('abc', indices_c, counts_abc[indices_c[0]])
print(sum(counts_abc))
for key, vals in adict_abc.items():
a = key
for val in vals:
b = val[1]
c = val[2]
for d in range(c + 1, mD+1):
mysum = sumsq(a,b,c,d)
if mysum in squares:
if a not in adict_abcd:
adict_abcd[a] = [[a,b,c,d]]
counts_abcd[a] += 1
else:
adict_abcd[a].append([a,b,c,d])
counts_abcd[a] += 1
mykey = str(a)+','+str(b)+','+str(c)+','+str(d)
sumsquares[mykey] = mysum
maxcountabcd = max(counts_abcd)
indices_d = [i for i, x in enumerate(counts_abcd) if x == maxcountabcd]
print('abcd', indices_d, counts_abcd[indices_d[0]])
print(sum(counts_abcd))
-------- output -------(edited to add the number found)
ab [240, 420, 840] (there was a tie each with 17 of them)
2673
abc [288] (59 of them)
5185
abcd [288] (121 of them)
7015
Edited on April 16, 2025, 1:01 pm
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Posted by Larry
on 2025-04-16 12:58:45 |
The code I used
