Euler's sum of powers conjecture: Difference between revisions

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Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769. It states that for all integers n and k greater than 1, if the sum of n kth powers of positive integers is itself a kth power, then n is greater than or equal to k.

In symbols, if $\sum_{i = 1}^{n} a_{i}^{k} = b^{k}$ where $n > 1$ and $a_{1}, a_{2}, \dots, a_{n}, b$ are positive integers, then $n \geq k$ .

The conjecture represents an attempt to generalize Fermat's last theorem, which could be seen as the special case of n = 2: if $a_{1}^{k} + a_{2}^{k} = b^{k}$ , then $2 \geq k$ .

Although the conjecture holds for the case of k = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for k = 4 and k = 5. It still remains unknown if the conjecture fails or holds for any value k ≥ 6.

Background

Euler had an equality for four fourth powers $5 9^{4} + 15 8^{4} = 13 3^{4} + 13 4^{4}$ , this however is not a counterexample. He also provided a complete solution to the four cubes problem as in Plato's number $3^{3} + 4^{3} + 5^{3} = 6^{3}$ or the taxicab number 1729.^[1]^[2] The general solution for:

x_{1}^{3} + x_{2}^{3} = x_{3}^{3} + x_{4}^{3}

is

x_{1} = 1 - (a - 3 b) (a^{2} + 3 b^{2}), x_{2} = (a + 3 b) (a^{2} + 3 b^{2}) - 1

x_{3} = (a + 3 b) - (a^{2} + 3 b^{2})^{2}, x_{4} = (a^{2} + 3 b^{2})^{2} - (a - 3 b)

where $a$ and $b$ are any integers.

Counterexamples

k = 5

The conjecture was disproven by L. J. Lander and T. R. Parkin in 1966 when, through a direct computer search on a CDC 6600, they found the following counterexample for k = 5:^[3]

27⁵ + 84⁵ + 110⁵ + 133⁵ = 144⁵.

Yet another counterexample 85282⁵ + 28969⁵ + 3183⁵ + 55⁵ = 85359⁵ was found by Jim Frye in 2004.

k = 4

In 1986, Noam Elkies found a method to construct an infinite series of counterexamples for the k = 4 case.^[4] His smallest counterexample was the following:

2682440⁴ + 15365639⁴ + 18796760⁴ = 20615673⁴.

A particular case of Elkies' solutions can be reduced to the identity:^[5]^[6]

(85v²+484v−313)⁴ + (68v²−586v+10)⁴ + (2u)⁴ = (357v²−204v+363)⁴

where

u² = 22030+28849v−56158v²+36941v³−31790v⁴.

This is an elliptic curve with a rational point with v₁ = −31/467. From this initial rational point, one can further compute an infinite number of v_i. Substituting v₁ into the identity and removing common factors gives the numerical example cited above.

In 1988, Roger Frye subsequently found the smallest possible counterexample for k = 4 by a direct computer search using techniquesPotter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park. suggested by Elkies:

95800⁴ + 217519⁴ + 414560⁴ = 422481⁴.

Moreover, this solution is the only one with values of the variables below 1,000,000.

Generalizations

In 1967, L. J. Lander, T. R. Parkin, and John Selfridge conjectured^[7] that if $\sum_{i = 1}^{n} a_{i}^{k} = \sum_{j = 1}^{m} b_{j}^{k}$ , where a_i ≠ b_j are positive integers for all 1 ≤ i ≤ n and 1 ≤ j ≤ m, then m+n ≥ k. This would imply as a special case that if

\sum_{i = 1}^{n} a_{i}^{k} = b^{k}

(under the conditions given above) then n ≥ k−1.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

External links

Tito Piezas III: A Collection of Algebraic Identities
EulerNet: Computing Minimal Equal Sums Of Like Powers
Jaroslaw Wroblewski Equal Sums of Like Powers
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Euler's Conjecture at library.thinkquest.org
A simple explanation of Euler's Conjecture at Maths Is Good For You!
Mathematicians Find New Solutions To An Ancient Puzzle
Ed Pegg Jr. Power Sums, Math Games

↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

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A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ Template:Cite web
↑ Template:Cite web
↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[1] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[2] Template:Cite web

[3] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[4] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[5] Template:Cite web

[6] Template:Cite web

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-== . Now in the third year Longchamp Pliage ==
+'''Euler's conjecture''' is a disproved [[conjecture]] in [[mathematics]] related to [[Fermat's last theorem]] which was proposed by [[Leonhard Euler]] in 1769. It states that for all [[integers]] ''n'' and ''k'' greater than 1, if the sum of ''n'' ''k''th powers of positive integers is itself a ''k''th power, then ''n'' is greater than or equal to ''k''.
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+In symbols, if
+<math>
-   <li>[http://www.vc1000.com/news/html/?83497.html http://www.vc1000.com/news/html/?83497.html]</li>
+\sum_{i=1}^{n} a_i^k = b^k
+</math>
-   <li>[http://111.11.181.205:8082/news/html/?55390.html http://111.11.181.205:8082/news/html/?55390.html]</li>
+where <math>n>1</math> and <math>a_1, a_2, \dots, a_n, b</math> are positive integers, then <math>n\geq k</math>.
-   <li>[http://enseignement-lsf.com/spip.php?article64#forum18145805 http://enseignement-lsf.com/spip.php?article64#forum18145805]</li>
-   <li>[http://online2games.net/activity/p/42936/ http://online2games.net/activity/p/42936/]</li>
- </ul>
-== the largest entity in the group Isabel Marant Brisbane ==
+The conjecture represents an attempt to generalize [[Fermat's last theorem]], which could be seen as the special case of ''n''&nbsp;=&nbsp;2: if <math>a_1^k + a_2^k = b^k</math>, then <math>2 \geq k</math>.
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+Although the conjecture holds for the case of ''k'' = 3 (which follows from Fermat's last theorem for the third powers), it was disproved for ''k'' = 4 and ''k'' = 5. It still remains unknown if the conjecture fails or holds for any value ''k'' ≥ 6.
-   <li>[http://neoplanning.net/demo/modules/d3blog/details.php?bid=8/ http://neoplanning.net/demo/modules/d3blog/details.php?bid=8/]</li>
+== Background ==
+Euler had an equality for four fourth powers <math>59^4 + 158^4 = 133^4 + 134^4</math>, this however is not a counterexample. He also provided a complete solution to the four cubes problem as in [[Plato's number]] <math>3^3+4^3+5^3=6^3</math> or the [[taxicab number]] 1729.<ref>{{cite book |title=The Genius of Euler: Reflections on His Life and Work |url=http://books.google.com/books?id=M4-zUnrSxNoC&pg=PA220 |editor=William Dunham |publisher= The MAA|page=220 |year=2007 |isbn=978-0-88385-558-4}}</ref><ref>{{cite web| url=http://www.oocities.org/titus_piezas/Equalsums.htm |title=Euler's Extended Conjecture |author=Titus Piezas III |year=2005}}</ref> The general solution for:
-   <li>[http://primeraventa.com/index.php?page=item&id=157214 http://primeraventa.com/index.php?page=item&id=157214]</li>
+:<math>x_1^3+x_2^3=x_3^3+x_4^3</math>
+is
-   <li>[http://www.jy250.com/forum.php?mod=viewthread&tid=40720&fromuid=8634 http://www.jy250.com/forum.php?mod=viewthread&tid=40720&fromuid=8634]</li>
+:<math>x_1 = 1-(a-3b)(a^2+3b^2), x_2 = (a+3b)(a^2+3b^2)-1</math>
+:<math>x_3 = (a+3b)-(a^2+3b^2)^2, x_4 = (a^2+3b^2)^2-(a-3b)</math>
-   <li>[http://blhsalumni.com/forum/read.php?1,12134 http://blhsalumni.com/forum/read.php?1,12134]</li>
+where <math>a</math> and <math>b</math> are any integers.
- </ul>
+== Counterexamples ==
+=== ''k'' = 5 ===
+The conjecture was disproven by [[Leon J. Lander|L. J. Lander]] and [[Thomas Parkin|T. R. Parkin]] in [[1966 in science|1966]] when, through a direct computer search on a [[CDC 6600]], they found the following counterexample for ''k'' = 5:<ref>{{cite journal |author=L. J. Lander, T. R. Parkin |title=Counterexample to Euler's conjecture on sums of like powers |journal=Bull. Amer. Math. Soc. |volume=72 |year=1966 |pages=1079 |doi=10.1090/S0002-9904-1966-11654-3}}</ref>
+::27<sup>5</sup> + 84<sup>5</sup> + 110<sup>5</sup> + 133<sup>5</sup> = [[144 (number)|144]]<sup>5</sup>.
+Yet another counterexample 85282<sup>5</sup> + 28969<sup>5</sup> + 3183<sup>5</sup> + 55<sup>5</sup> = 85359<sup>5</sup> was found by [[Jim Frye]] in 2004.
+=== ''k'' = 4 ===
+In 1986, [[Noam Elkies]] found a method to construct an infinite series of counterexamples for the ''k'' = 4 case.<ref>{{cite journal |author=Noam Elkies |title=On ''A''<sup>4</sup> + ''B''<sup>4</sup> + ''C''<sup>4</sup> = ''D''<sup>4</sup> |journal=[[Mathematics of Computation]] |year=1988 |volume=51 |issue=184 |pages=825–835 |doi=10.2307/2008781 |mr=0930224 |jstor=2008781}}</ref> His smallest counterexample was the following:
+::2682440<sup>4</sup> + 15365639<sup>4</sup> + 18796760<sup>4</sup> = 20615673<sup>4</sup>.
+A particular case of Elkies' solutions can be reduced to the identity:<ref>{{cite web|url=http://groups.google.com/group/sci.math/browse_thread/thread/15beef75eaddcb1b?hl=en#|title=Elkies' a^4+b^4+c^4 = d^4}}</ref><ref>{{cite web|url=http://sites.google.com/site/tpiezas/014|title=Sums of Three Fourth Powers}}</ref>
+::(85''v''<sup>2</sup>+484''v''−313)<sup>4</sup> + (68''v''<sup>2</sup>−586''v''+10)<sup>4</sup> + (2''u'')<sup>4</sup> = (357''v''<sup>2</sup>−204''v''+363)<sup>4</sup>
+where
+::''u''<sup>2</sup> = 22030+28849''v''−56158''v''<sup>2</sup>+36941''v''<sup>3</sup>−31790''v''<sup>4</sup>.
+This is an [[elliptic curve]] with a [[rational point]] with ''v''<sub>1</sub> = −31/467. From this initial rational point, one can further compute an infinite number of ''v<sub>i</sub>''. Substituting ''v''<sub>1</sub> into the identity and removing common factors gives the numerical example cited above.
+In 1988, [[Roger Frye]] subsequently found the smallest possible counterexample for ''k'' = 4 by a direct computer search using techniques{{Citation needed|date=May 2009}} suggested by Elkies:
+::95800<sup>4</sup> + 217519<sup>4</sup> + 414560<sup>4</sup> = 422481<sup>4</sup>.
+Moreover, this solution is the only one with values of the variables below 1,000,000.
+== Generalizations ==
+In 1967, L. J. Lander, T. R. Parkin, and [[John Selfridge]] conjectured<ref>{{cite journal |author=L. J. Lander, T. R. Parkin, J. L. Selfridge |title=A Survey of Equal Sums of Like Powers |journal=[[Mathematics of Computation]] |volume=21 |issue=99 |year=1967 |pages=446–459 |doi=10.1090/S0025-5718-1967-0222008-0 |jstor=2003249}}</ref> that if <math>\sum_{i=1}^{n} a_i^k = \sum_{j=1}^{m} b_j^k</math>, where ''a<sub>i</sub>''&nbsp;≠&nbsp;''b<sub>j</sub>'' are positive integers for all 1&nbsp;≤&nbsp;''i''&nbsp;≤&nbsp;''n'' and 1&nbsp;≤&nbsp;''j''&nbsp;≤&nbsp;''m'', then ''m''+''n''&nbsp;≥&nbsp;''k''.
+This would imply as a special case that if
+::<math>\sum_{i=1}^{n} a_i^k = b^k</math>
+(under the conditions given above) then ''n''&nbsp;≥&nbsp;''k''−1.
+==See also==
+*[[Lander, Parkin, and Selfridge conjecture]]
+* [[Jacobi–Madden equation]]
+*[[Prouhet–Tarry–Escott problem]]
+*[[Beal's conjecture]]
+*[[Pythagorean quadruple]]
+*[[Sums of powers]], a list of related conjectures and theorems
+== References ==
+{{reflist}}
+== External links ==
+* Tito Piezas III: [http://sites.google.com/site/tpiezas/Home/  A Collection of Algebraic Identities]
+* [http://euler.free.fr/ EulerNet: Computing Minimal Equal Sums Of Like Powers]
+* Jaroslaw Wroblewski [http://www.math.uni.wroc.pl/~jwr/eslp/  Equal Sums of Like Powers]
+* {{MathWorld |title=Euler's Sum of Powers Conjecture |urlname=EulersSumofPowersConjecture}}
+* {{MathWorld |title=Euler Quartic Conjecture |urlname=EulerQuarticConjecture}}
+* {{MathWorld |title=Diophantine Equation--4th Powers |urlname=DiophantineEquation4thPowers}}
+* [http://library.thinkquest.org/28049/Euler's%20conjecture.html Euler's Conjecture] at library.thinkquest.org
+* [http://www.mathsisgoodforyou.com/conjecturestheorems/eulerconjecture.htm A simple explanation of Euler's Conjecture] at Maths Is Good For You!
+* [http://www.sciencedaily.com/releases/2008/03/080314145039.htm Mathematicians Find New Solutions To An Ancient Puzzle]
+* Ed Pegg Jr. [http://www.maa.org/editorial/mathgames/mathgames_11_13_06.html Power Sums], Math Games
+[[Category:Number theory]]
+[[Category:Diophantine equations]]
+[[Category:Disproved conjectures]]

Euler's sum of powers conjecture: Difference between revisions

Revision as of 00:43, 24 January 2014

Contents

Background

Counterexamples

k = 5

k = 4

Generalizations

See also

References

External links

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Euler's sum of powers conjecture: Difference between revisions

Revision as of 00:43, 24 January 2014

Background

Counterexamples

k = 5

k = 4

Generalizations

See also

References

External links

Navigation menu

Search