# 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
2 years, 11 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.

- 1 year, 5 months ago

- 2 years, 9 months ago

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?

- 2 years, 10 months ago

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 # -*- 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]) 

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 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 

- 2 years, 11 months ago

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 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 

- 2 years, 11 months ago

If we stipulate that both sides have to have at least two terms, then I get some solutions: \begin{aligned} 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{aligned} And there are lots more.

- 2 years, 11 months ago

How u got these?

- 2 years, 11 months ago

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 $z-w \le 1.$

- 2 years, 11 months ago

Woah!! :P

- 2 years, 11 months ago