Squares = Cubes?

\[1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2 = 3^3 + 4^3 = 91.\]

Are there any other (non-trivial) sets of consecutive squares and cubes whose sums are equal?

Note by Eli Ross
1 year, 6 months ago

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If we stipulate that both sides have to have at least two terms, then I get some solutions: \[ \begin{align} 6^2 + \cdots + 54^2 &= 20^3 + \cdots + 24^3 \\ 7^2 + \cdots + 12^2 &= 6^3 + 7^3 \\ 8^2 + \cdots + 57^2 &= 8^3 + \cdots + 22^3 \\ 11^2 + \cdots + 15^2 &= 7^3 + 8^3 \\ 14^2 + \cdots + 54^2 &= 6^3 + \cdots + 21^3 \\ 15^2 + \cdots + 91^2 &= 30^3 + \cdots + 36^3 \\ 20^2 + \cdots + 30^2 &= 9^3 + \cdots + 13^3 \\ 22^2 + \cdots + 94^2 &= 31^3 + \cdots + 37^3 \\ 22^2 + \cdots + 96^2 &= 17^3 + \cdots + 33^3 \\ 27^2 + \cdots + 59^2 &= 1^3 + \cdots + 22^3 \\ \end{align} \] And there are lots more.

Patrick Corn - 1 year, 6 months ago

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How u got these?

Md Zuhair - 1 year, 6 months ago

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By a (very unsophisticated) search for positive integer solutions to the polynomial equation \[ \frac{x(x+1)(2x+1)}6 - \frac{y(y+1)(2y+1)}6 = \frac{z^2(z+1)^2}4 - \frac{w^2(w+1)^2}4, \] where I throw out solutions with \(x-y \le 1\) or \(z-w \le 1.\)

Patrick Corn - 1 year, 6 months ago

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@Patrick Corn Woah!! :P

Md Zuhair - 1 year, 6 months ago

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# -*- coding: utf-8 -*-
"""
Created on Wed Nov 22 16:41:38 2017

@author: Michael Fitzgerald
"""

def build_dict(n, s, sum_x):
    for i in range(max_, 1,-1):
        for j in range(min_, i):
            for k in range(j,i+1):
                if i <= j:
                    break
                if j != k:
                    key_ = '%d-%d' % (j, k)
                else:
                    key_ = '%d' % k
                s += k**n
                sum_x[key_] = s
            s = 0
    return sum_x

min_ = 1   #Enter min of range
max_ = 300   #Enter max of range

sum_sq = {}
sum_ = 0
sum_sq = build_dict(2,sum_, sum_sq)

sum_cube = {}
sum_ = 0
sum_cube = build_dict(3, sum_, sum_cube)

#print sum_cube

matching = [[sum_sq[a],a, b] for a in sum_sq for b in sum_cube if sum_sq[a] == sum_cube[b]]
sorted_list = sorted(matching, key = lambda x: x[0])
for i in sorted_list:
    print 'Sum: %d; Squares range: %s; Cubes range: %s' %  (i[0], i[1], i[2])   

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Sum: 1; Squares range: 1; Cubes range: 1
Sum: 9; Squares range: 3; Cubes range: 1-2
Sum: 36; Squares range: 6; Cubes range: 1-3
Sum: 64; Squares range: 8; Cubes range: 4
Sum: 91; Squares range: 1-6; Cubes range: 3-4
Sum: 100; Squares range: 10; Cubes range: 1-4
Sum: 225; Squares range: 15; Cubes range: 1-5
Sum: 441; Squares range: 21; Cubes range: 1-6
Sum: 559; Squares range: 7-12; Cubes range: 6-7
Sum: 729; Squares range: 27; Cubes range: 9
Sum: 784; Squares range: 28; Cubes range: 1-7
Sum: 855; Squares range: 11-15; Cubes range: 7-8
Sum: 1296; Squares range: 36; Cubes range: 1-8
Sum: 2025; Squares range: 45; Cubes range: 1-9
Sum: 3025; Squares range: 55; Cubes range: 1-10
Sum: 4096; Squares range: 64; Cubes range: 16
Sum: 4356; Squares range: 66; Cubes range: 1-11
Sum: 6084; Squares range: 78; Cubes range: 1-12
Sum: 6985; Squares range: 20-30; Cubes range: 9-13
Sum: 8281; Squares range: 91; Cubes range: 1-13
Sum: 11025; Squares range: 105; Cubes range: 1-14
Sum: 14400; Squares range: 120; Cubes range: 1-15
Sum: 15625; Squares range: 125; Cubes range: 25
Sum: 18496; Squares range: 136; Cubes range: 1-16
Sum: 23409; Squares range: 153; Cubes range: 1-17
Sum: 29240; Squares range: 35-50; Cubes range: 2-18
Sum: 29241; Squares range: 171; Cubes range: 1-18
Sum: 36100; Squares range: 190; Cubes range: 1-19
Sum: 41616; Squares range: 204; Cubes range: 23-25
Sum: 44100; Squares range: 210; Cubes range: 1-20
Sum: 46656; Squares range: 216; Cubes range: 36
Sum: 47025; Squares range: 28-54; Cubes range: 24-26
Sum: 53136; Squares range: 14-54; Cubes range: 6-21
Sum: 53361; Squares range: 231; Cubes range: 1-21
Sum: 53900; Squares range: 6-54; Cubes range: 20-24
Sum: 63225; Squares range: 8-57; Cubes range: 8-22
Sum: 64009; Squares range: 253; Cubes range: 1-22
Sum: 64009; Squares range: 27-59; Cubes range: 1-22
Sum: 76175; Squares range: 48-69; Cubes range: 2-23
Sum: 76176; Squares range: 276; Cubes range: 1-23
Sum: 89559; Squares range: 38-68; Cubes range: 7-24
Sum: 89559; Squares range: 38-68; Cubes range: 30-32
Sum: 103823; Squares range: 22-68; Cubes range: 47
Sum: 108801; Squares range: 50-76; Cubes range: 16-26
Sum: 186200; Squares range: 36-84; Cubes range: 11-29
Sum: 186200; Squares range: 74-98; Cubes range: 11-29
Sum: 245575; Squares range: 48-94; Cubes range: 7-31
Sum: 246015; Squares range: 126-139; Cubes range: 2-31
Sum: 254331; Squares range: 15-91; Cubes range: 30-36
Sum: 274625; Squares range: 90-115; Cubes range: 65
Sum: 277984; Squares range: 22-94; Cubes range: 31-37
Sum: 296225; Squares range: 22-96; Cubes range: 17-33
Sum: 300321; Squares range: 53-101; Cubes range: 16-33
Sum: 339625; Squares range: 85-117; Cubes range: 16-34
Sum: 404209; Squares range: 52-110; Cubes range: 25-37
Sum: 461384; Squares range: 72-120; Cubes range: 35-42
Sum: 485199; Squares range: 67-120; Cubes range: 27-39
Sum: 643159; Squares range: 7-124; Cubes range: 19-40
Sum: 741320; Squares range: 100-147; Cubes range: 2-41
Sum: 750519; Squares range: 225-238; Cubes range: 62-64
Sum: 810216; Squares range: 14-134; Cubes range: 43-50
Sum: 815309; Squares range: 77-142; Cubes range: 5-42
Sum: 825209; Squares range: 105-153; Cubes range: 32-45
Sum: 841555; Squares range: 27-136; Cubes range: 22-43
Sum: 890560; Squares range: 148-180; Cubes range: 12-43
Sum: 1036664; Squares range: 247-262; Cubes range: 47-54
Sum: 1083475; Squares range: 150-187; Cubes range: 43-52
Sum: 1092385; Squares range: 106-164; Cubes range: 24-46
Sum: 1382975; Squares range: 148-194; Cubes range: 2-48
Sum: 1424124; Squares range: 27-162; Cubes range: 77-79
Sum: 1442609; Squares range: 147-195; Cubes range: 56-62
Sum: 1494541; Squares range: 104-177; Cubes range: 13-49
Sum: 1739780; Squares range: 46-174; Cubes range: 17-51
Sum: 1752309; Squares range: 247-272; Cubes range: 45-57
Sum: 1779184; Squares range: 216-248; Cubes range: 57-64
Sum: 1801745; Squares range: 6-175; Cubes range: 32-53
Sum: 1831536; Squares range: 16-176; Cubes range: 31-53
Sum: 1854784; Squares range: 122-194; Cubes range: 21-52
Sum: 1926441; Squares range: 254-280; Cubes range: 33-54
Sum: 2030301; Squares range: 140-206; Cubes range: 100-101
Sum: 2197160; Squares range: 190-237; Cubes range: 74-78
Sum: 2906541; Squares range: 91-211; Cubes range: 76-81
Sum: 3102884; Squares range: 27-210; Cubes range: 32-60
Sum: 3328641; Squares range: 55-216; Cubes range: 48-65
Sum: 3374225; Squares range: 174-248; Cubes range: 65-74
Sum: 3455955; Squares range: 226-279; Cubes range: 54-68
Sum: 3770585; Squares range: 15-224; Cubes range: 89-93
Sum: 3848031; Squares range: 154-247; Cubes range: 31-63
Sum: 4063815; Squares range: 158-252; Cubes range: 7-63
Sum: 4073949; Squares range: 236-293; Cubes range: 52-69
Sum: 4329369; Squares range: 137-249; Cubes range: 112-114
Sum: 4837456; Squares range: 78-246; Cubes range: 81-88
Sum: 5110784; Squares range: 92-252; Cubes range: 107-110
Sum: 5149691; Squares range: 44-249; Cubes range: 35-68
Sum: 5184720; Squares range: 185-279; Cubes range: 119-121
Sum: 5338880; Squares range: 136-264; Cubes range: 29-68
Sum: 5370400; Squares range: 126-262; Cubes range: 84-91
Sum: 5492691; Squares range: 209-294; Cubes range: 15-68
Sum: 5813729; Squares range: 145-273; Cubes range: 17-69
Sum: 6175000; Squares range: 58-265; Cubes range: 6-70
Sum: 6953939; Squares range: 107-280; Cubes range: 47-75 

Michael Fitzgerald - 1 year, 6 months ago

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Sum: 1; Squares range: 1; Tesseract (^4) range: 1
Sum: 16; Squares range: 4; Tesseract (^4) range: 2
Sum: 81; Squares range: 9; Tesseract (^4) range: 3
Sum: 256; Squares range: 16; Tesseract (^4) range: 4
Sum: 625; Squares range: 25; Tesseract (^4) range: 5
Sum: 1296; Squares range: 36; Tesseract (^4) range: 6
Sum: 2401; Squares range: 49; Tesseract (^4) range: 7
Sum: 4096; Squares range: 64; Tesseract (^4) range: 8
Sum: 6561; Squares range: 81; Tesseract (^4) range: 9
Sum: 10000; Squares range: 100; Tesseract (^4) range: 10
Sum: 14641; Squares range: 121; Tesseract (^4) range: 11
Sum: 20736; Squares range: 144; Tesseract (^4) range: 12
Sum: 24979; Squares range: 62-67; Tesseract (^4) range: 5-10
Sum: 28561; Squares range: 119-120; Tesseract (^4) range: 13
Sum: 28561; Squares range: 169; Tesseract (^4) range: 13
Sum: 38416; Squares range: 196; Tesseract (^4) range: 14
Sum: 50625; Squares range: 225; Tesseract (^4) range: 15
Sum: 59731; Squares range: 11-56; Tesseract (^4) range: 6-12
Sum: 65536; Squares range: 256; Tesseract (^4) range: 16
Sum: 83521; Squares range: 289; Tesseract (^4) range: 17
Sum: 218515; Squares range: 34-88; Tesseract (^4) range: 11-16
Sum: 562666; Squares range: 79-129; Tesseract (^4) range: 1-19
Sum: 562666; Squares range: 163-181; Tesseract (^4) range: 1-19
Sum: 722665; Squares range: 16-129; Tesseract (^4) range: 2-20
Sum: 907555; Squares range: 56-142; Tesseract (^4) range: 17-22
Sum: 1590979; Squares range: 163-208; Tesseract (^4) range: 20-25
Sum: 2153291; Squares range: 155-216; Tesseract (^4) range: 5-25

Michael Fitzgerald - 1 year, 6 months ago

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The difference of the squares of two consecutive triangular numbers will be a perfect cube whose cube root is equal to the difference between those triangular numbers. So, 3^3 + 4^3 = (6^2 - 3^2) + (10^2 - 6^2) = 91 = 1^2 + 2^2 + 3^2 + 4^2 + 5^2 + 6^2. Or generalized, the sum of the cubes of two consecutive numbers is obtained; (L-M)^3 + [(L-M)+1]^3 = (L^2 - M^2) + (N^2 - O^2) where n-o = [(L-M)+1] and (N-O)-1 = L-M. Does that help at all?

Hiro Rumpf - 1 year, 5 months ago

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Sowmya Surapaneni - 1 year, 4 months ago

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I'm curious about these things, but are they coincidental or are there deeper algebraic reasons for this? Say rooted in abstract algebra or number theoretical grounds? For example, Fermat's Last Theorem looked simple when stated, but the proof required techniques deemed not available during Fermat's time, but somehow involved elliptic curves and such things rooted in abstract algebra.

Max Yuen - 3 weeks ago

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