Counterexamples to Euler's sum of powers conjecture

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Numbers

$k$

$b$

$a_i$ or $b$

$95800^4+217519^4+414560^4=422481^4$

$a_1$:

95800

$95800^4+217519^4+414560^4=422481^4$

$a_2$:

217519

$95800^4+217519^4+414560^4=422481^4$

$a_3$:

414560

$95800^4+217519^4+414560^4=422481^4$

$b$:

422481

$673865^4+1390400^4+2767624^4=2813001^4$

$a_1$:

673865

$673865^4+1390400^4+2767624^4=2813001^4$

$a_2$:

1390400

$673865^4+1390400^4+2767624^4=2813001^4$

$a_3$:

2767624

$673865^4+1390400^4+2767624^4=2813001^4$

$b$:

2813001

$1705575^4+5507880^4+8332208^4=8707481^4$

$a_1$:

1705575

$1705575^4+5507880^4+8332208^4=8707481^4$

$a_2$:

5507880

$1705575^4+5507880^4+8332208^4=8707481^4$

$a_3$:

8332208

$1705575^4+5507880^4+8332208^4=8707481^4$

$b$:

8707481

$5870000^4+8282543^4+11289040^4=12197457^4$

$a_1$:

5870000

$5870000^4+8282543^4+11289040^4=12197457^4$

$a_2$:

8282543

$5870000^4+8282543^4+11289040^4=12197457^4$

$a_3$:

11289040

$5870000^4+8282543^4+11289040^4=12197457^4$

$b$:

12197457

$4479031^4+12552200^4+14173720^4=16003017^4$

$a_1$:

4479031

$4479031^4+12552200^4+14173720^4=16003017^4$

$a_2$:

12552200

$4479031^4+12552200^4+14173720^4=16003017^4$

$a_3$:

14173720

$4479031^4+12552200^4+14173720^4=16003017^4$

$b$:

16003017

$3642840^4+7028600^4+16281009^4=16430513^4$

$a_1$:

3642840

$3642840^4+7028600^4+16281009^4=16430513^4$

$a_2$:

7028600

$3642840^4+7028600^4+16281009^4=16430513^4$

$a_3$:

16281009

$3642840^4+7028600^4+16281009^4=16430513^4$

$b$:

16430513

$2682440^4+15365639^4+18796760^4=20615673^4$

$a_1$:

2682440

$2682440^4+15365639^4+18796760^4=20615673^4$

$a_2$:

15365639

$2682440^4+15365639^4+18796760^4=20615673^4$

$a_3$:

18796760

$2682440^4+15365639^4+18796760^4=20615673^4$

$b$:

20615673

$2164632^4+31669120^4+41084175^4=44310257^4$

$a_1$:

2164632

$2164632^4+31669120^4+41084175^4=44310257^4$

$a_2$:

31669120

$2164632^4+31669120^4+41084175^4=44310257^4$

$a_3$:

41084175

$2164632^4+31669120^4+41084175^4=44310257^4$

$b$:

44310257

$10409096^4+42878560^4+65932985^4=68711097^4$

$a_1$:

10409096

$10409096^4+42878560^4+65932985^4=68711097^4$

$a_2$:

42878560

$10409096^4+42878560^4+65932985^4=68711097^4$

$a_3$:

65932985

$10409096^4+42878560^4+65932985^4=68711097^4$

$b$:

68711097

$34918520^4+87865617^4+106161120^4=117112081^4$

$a_1$:

34918520

$34918520^4+87865617^4+106161120^4=117112081^4$

$a_2$:

87865617

$34918520^4+87865617^4+106161120^4=117112081^4$

$a_3$:

106161120

$34918520^4+87865617^4+106161120^4=117112081^4$

$b$:

117112081

$1841160^4+121952168^4+122055375^4=145087793^4$

$a_1$:

1841160

$1841160^4+121952168^4+122055375^4=145087793^4$

$a_2$:

121952168

$1841160^4+121952168^4+122055375^4=145087793^4$

$a_3$:

122055375

$1841160^4+121952168^4+122055375^4=145087793^4$

$b$:

145087793

$27450160^4+108644015^4+146627384^4=156646737^4$

$a_1$:

27450160

$27450160^4+108644015^4+146627384^4=156646737^4$

$a_2$:

108644015

$27450160^4+108644015^4+146627384^4=156646737^4$

$a_3$:

146627384

$27450160^4+108644015^4+146627384^4=156646737^4$

$b$:

156646737

$186668000^4+260052385^4+582665296^4=589845921^4$

$a_1$:

186668000

$186668000^4+260052385^4+582665296^4=589845921^4$

$a_2$:

260052385

$186668000^4+260052385^4+582665296^4=589845921^4$

$a_3$:

582665296

$186668000^4+260052385^4+582665296^4=589845921^4$

$b$:

589845921

$219076465^4+275156240^4+630662624^4=638523249^4$

$a_1$:

219076465

$219076465^4+275156240^4+630662624^4=638523249^4$

$a_2$:

275156240

$219076465^4+275156240^4+630662624^4=638523249^4$

$a_3$:

630662624

$219076465^4+275156240^4+630662624^4=638523249^4$

$b$:

638523249

$558424440^4+606710871^4+769321280^4=873822121^4$

$a_1$:

558424440

$558424440^4+606710871^4+769321280^4=873822121^4$

$a_2$:

606710871

$558424440^4+606710871^4+769321280^4=873822121^4$

$a_3$:

769321280

$558424440^4+606710871^4+769321280^4=873822121^4$

$b$:

873822121

$588903336^4+859396455^4+1166705840^4=1259768473^4$

$a_1$:

588903336

$588903336^4+859396455^4+1166705840^4=1259768473^4$

$a_2$:

859396455

$588903336^4+859396455^4+1166705840^4=1259768473^4$

$a_3$:

1166705840

$588903336^4+859396455^4+1166705840^4=1259768473^4$

$b$:

1259768473

$50237800^4+632671960^4+1670617271^4=1679142729^4$

$a_1$:

50237800

$50237800^4+632671960^4+1670617271^4=1679142729^4$

$a_2$:

632671960

$50237800^4+632671960^4+1670617271^4=1679142729^4$

$a_3$:

1670617271

$50237800^4+632671960^4+1670617271^4=1679142729^4$

$b$:

1679142729

$686398000^4+1237796960^4+1662997663^4=1787882337^4$

$a_1$:

686398000

$686398000^4+1237796960^4+1662997663^4=1787882337^4$

$a_2$:

1237796960

$686398000^4+1237796960^4+1662997663^4=1787882337^4$

$a_3$:

1662997663

$686398000^4+1237796960^4+1662997663^4=1787882337^4$

$b$:

1787882337

$92622401^4+1553556440^4+1593513080^4=1871713857^4$

$a_1$:

92622401

$92622401^4+1553556440^4+1593513080^4=1871713857^4$

$a_2$:

1553556440

$92622401^4+1553556440^4+1593513080^4=1871713857^4$

$a_3$:

1593513080

$92622401^4+1553556440^4+1593513080^4=1871713857^4$

$b$:

1871713857

$27^5+84^5+110^5+133^5=144^5$

$a_1$:

$27^5+84^5+110^5+133^5=144^5$

$a_2$:

$27^5+84^5+110^5+133^5=144^5$

$a_3$:

110

$27^5+84^5+110^5+133^5=144^5$

$a_4$:

133

$27^5+84^5+110^5+133^5=144^5$

$b$:

144

$(-220)^5+5027^5+6237^5+14068^5=14132^5$

$a_1$:

-220

$(-220)^5+5027^5+6237^5+14068^5=14132^5$

$a_2$:

5027

$(-220)^5+5027^5+6237^5+14068^5=14132^5$

$a_3$:

6237

$(-220)^5+5027^5+6237^5+14068^5=14132^5$

$a_4$:

14068

$(-220)^5+5027^5+6237^5+14068^5=14132^5$

$b$:

14132

$55^5+3183^5+28969^5+85282^5=85359^5$

$a_1$:

$55^5+3183^5+28969^5+85282^5=85359^5$

$a_2$:

3183

$55^5+3183^5+28969^5+85282^5=85359^5$

$a_3$:

28969

$55^5+3183^5+28969^5+85282^5=85359^5$

$a_4$:

85282

$55^5+3183^5+28969^5+85282^5=85359^5$

$b$:

85359

$(-1340632)^5+719115^5+1331622^5+1956213^5=1956878^5$

$a_1$:

-1340632

$(-1340632)^5+719115^5+1331622^5+1956213^5=1956878^5$

$a_2$:

719115

$(-1340632)^5+719115^5+1331622^5+1956213^5=1956878^5$

$a_3$:

1331622

$(-1340632)^5+719115^5+1331622^5+1956213^5=1956878^5$

$a_4$:

1956213

$(-1340632)^5+719115^5+1331622^5+1956213^5=1956878^5$

$b$:

1956878

Definition

Primitive nonzero integer solutions of $a_1^k+\cdots+a_{k-1}^k=b^k$ with $b>|a_i|$ for every $i$ and no pair $a_i=-a_j$; for even $k$ the $a_i$ are positive. This table stores the parts $a_i$ and $b$. Rows with all $a_i>0$ are counterexamples to Euler's sum of powers conjecture [4].

Parameters

$k$

— exponent (an integer exponent $k>2$)

$b$

— solution index (the largest term $b$ in a stored solution, so $b>0$ and $b>|a_i|$ for every summand)

Comments

(1)

The fifth-power rows follow [4] and [1] in allowing integer summands. When one summand is negative, moving it to the other side gives an equal-sums-of-like-powers identity in positive terms, with five terms across the two sides; those signed rows are not counterexamples to Euler's conjecture with one fifth power equal to a sum of four positive fifth powers.

(2)

The summands $a_i$ are ordered by integer value. The fourth-power rows have no $a_4$ part because they have three summands.

(3)

The row $b=422481$ is the smallest fourth-power solution, found by Frye [3], and $b=20615673$ is the first solution Elkies found [2].

References

[1]

Jeffrey M. Braun, The fourth known primitive solution to $a^5+b^5+c^5+d^5=e^5$, 2026. (arXiv)

[2]

Noam D. Elkies, On $A^4+B^4+C^4=D^4$, Mathematics of Computation 51 (1988), no. 184, 825-835. (doi)

[3]

Roger E. Frye, Finding $95800^4+217519^4+414560^4=422481^4$ on the Connection Machine, Proceedings of Supercomputing 88, Vol. II, Science and Applications, 1988, 106-116. (doi)

Links

[4]

Wikipedia: Euler's sum of powers conjecture

[5]

OEIS A003828

Similar tables

Known solutions of the Fermat-Catalan equation $x^p+y^q=z^r$ — records another family where one perfect power is a sum of other perfect powers

Good examples of Hall's conjecture — records small values of $|y^2-x^3|$, another measure of how close perfect powers come

Sums of powers $S_p(n)$ — stores sums over consecutive bases, where this table stores sparse bases in equal-sums identities

Data properties

Entries are of type: integer

How well the digits are known: exact

Table is complete: no (it holds the four fifth-power primitive solutions listed in [4] and [1], and the fourth-power solutions listed by OEIS A003828 [5], whose sequence entry says there are no further terms up to $1986560000$)

Sources of data: [4], [5], [1], [2], [3]

How they were obtained:

Every stored value is an exact integer. The generator checks each identity $a_1^k+\cdots+a_{k-1}^k=b^k$, the primitivity condition and the absence of cancelling pairs $a_i=-a_j$ by exact integer arithmetic.

It checks the fourth-power $b$ column against the OEIS A003828 b-file, and checks the $b=20615673$ row against Elkies' parametrisation at $v=-31/467$ after clearing denominators and common factors.