pourquoi y'a eu plein de mauvaises preuves pour le dernier théorème de Fermat
Prizes and incorrect proofs

In 1816, and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem. In 1857, the academy awarded 3,000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize. Another prize was offered in 1883 by the Academy of Brussels.

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks —a large sum at the time—to the Göttingen Academy of Sciences to offer as a prize for a complete proof of Fermat's Last Theorem. On 27 June 1908, the academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun. Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997. In March 2016, Wiles was awarded the Norwegian government's Abel prize worth €600,000 for "his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory".

Prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3.0 meters) of correspondence. In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. According to some claims, Edmund Landau tended to use a special preprinted form for such proofs, where the location of the first mistake was left blank to be filled by one of his graduate students. According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted by "people with a technical education but a failed career". In the words of mathematical historian Howard Eves, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."
Le dernier théorème de Fermat

En théorie des nombres, le dernier théorème de Fermat (parfois appelé conjecture de Fermat, surtout dans les textes anciens) énonce qu'il n'existe aucun ensemble de trois entiers positifs a, b et c qui puisse satisfaire l'équation aⁿ + bⁿ = cⁿ pour toute valeur entière de n supérieure à deux.

Ce théorème a été d'abord conjecturé par Pierre de Fermat en 1637 dans la marge d'une copie de l'Arithmétique, où il affirmait avoir une preuve qui était trop volumineuse pour tenir dans la marge. La première preuve réussie a été publiée en 1994 par Andrew Wiles, et formellement publiée en 1995, après 358 ans d'efforts des mathématiciens. Le problème non résolu a stimulé le développement de la théorie algébrique des nombres au XIXe siècle, et la preuve du théorème de modularité au XXe siècle. Il est l'un des théorèmes les plus notables de l'histoire des mathématiques, et avant sa preuve, il figurait dans le Livre Guinness des records pour « problème mathématique le plus difficile ».
### Query ###
pourquoi y'a eu plein de mauvaises preuves pour le dernier théorème de Fermat

### Main Text ###
Prizes and incorrect proofs

In 1816, and again in 1850, the French Academy of Sciences offered a prize for a general proof of Fermat's Last Theorem. In 1857, the academy awarded 3,000 francs and a gold medal to Kummer for his research on ideal numbers, although he had not submitted an entry for the prize. Another prize was offered in 1883 by the Academy of Brussels.

In 1908, the German industrialist and amateur mathematician Paul Wolfskehl bequeathed 100,000 gold marks —a large sum at the time—to the Göttingen Academy of Sciences to offer as a prize for a complete proof of Fermat's Last Theorem. On 27 June 1908, the academy published nine rules for awarding the prize. Among other things, these rules required that the proof be published in a peer-reviewed journal; the prize would not be awarded until two years after the publication; and that no prize would be given after 13 September 2007, roughly a century after the competition was begun. Wiles collected the Wolfskehl prize money, then worth $50,000, on 27 June 1997. In March 2016, Wiles was awarded the Norwegian government's Abel prize worth €600,000 for "his stunning proof of Fermat's Last Theorem by way of the modularity conjecture for semistable elliptic curves, opening a new era in number theory".

Prior to Wiles's proof, thousands of incorrect proofs were submitted to the Wolfskehl committee, amounting to roughly 10 feet (3.0 meters) of correspondence. In the first year alone (1907–1908), 621 attempted proofs were submitted, although by the 1970s, the rate of submission had decreased to roughly 3–4 attempted proofs per month. According to some claims, Edmund Landau tended to use a special preprinted form for such proofs, where the location of the first mistake was left blank to be filled by one of his graduate students. According to F. Schlichting, a Wolfskehl reviewer, most of the proofs were based on elementary methods taught in schools, and often submitted by "people with a technical education but a failed career". In the words of mathematical historian Howard Eves, "Fermat's Last Theorem has the peculiar distinction of being the mathematical problem for which the greatest number of incorrect proofs have been published."

### Supplementary Text ###
Le dernier théorème de Fermat

En théorie des nombres, le dernier théorème de Fermat (parfois appelé conjecture de Fermat, surtout dans les textes anciens) énonce qu'il n'existe aucun ensemble de trois entiers positifs a, b et c qui puisse satisfaire l'équation aⁿ + bⁿ = cⁿ pour toute valeur entière de n supérieure à deux.

Ce théorème a été d'abord conjecturé par Pierre de Fermat en 1637 dans la marge d'une copie de l'Arithmétique, où il affirmait avoir une preuve qui était trop volumineuse pour tenir dans la marge. La première preuve réussie a été publiée en 1994 par Andrew Wiles, et formellement publiée en 1995, après 358 ans d'efforts des mathématiciens. Le problème non résolu a stimulé le développement de la théorie algébrique des nombres au XIXe siècle, et la preuve du théorème de modularité au XXe siècle. Il est l'un des théorèmes les plus notables de l'histoire des mathématiques, et avant sa preuve, il figurait dans le Livre Guinness des records pour « problème mathématique le plus difficile ».

### Draft ###

Query: "pourquoi y'a eu plein de mauvaises preuves pour le dernier théorème de Fermat"
→ "why were there lots of bad proofs for Fermat's Last Theorem"

**Core issue**: User asking about abundance of incorrect solutions. Simple retrieval task but needs context.

### 1. What's the "dernier théorème"?
● Fermat's Last Theorem: no integer solutions to aⁿ + bⁿ = cⁿ when n>2
● Statement itself simple. Proof resisted ~350 years
● "Last" = final major theorem before full proof

### 2. Why so many failed attempts?

**Accessibility paradox**:
- Statement: elementary algebra
- Proof: required centuries of advanced mathematics
- Gap: thousands tried bridging impossible leap

**Historical factors**:
- Prize money incentives (esp. Wolfskehl 1908: 100,000 marks)
- Fame motivation  
- Mathematical tourism - amateurs seeing "easy" problem

### 3. Scale of failure

**Wolfskehl competition data** ●:
- 1907-1908: 621 proofs submitted first year
- Monthly rate: 3-4 submissions 1970s
- Total: thousands of failed attempts
- Physical evidence: ~10 feet correspondence, filling Göttingen archives

**Demographics** ◐:
- Technical education background
- Failed career academics/amateurs  
- Underestimation of required sophistication

### 4. Mathematical reality check

Elementary approaches doomed by:
- Modular forms theory
- Elliptic curves
- Galois representations  
- Taniyama-Shimura theorem (1960s)
- Iwasawa theory
- ∴ Centuries of mathematical development required

⟨H≈0.8⟩ The theorem became a mathematical Rorschach test - people saw elementary statement but projected their own mathematical sophistication onto it. Every failed proof revealed not logical flaw but gap in understanding of mathematical depth required. Like trying to explain quantum mechanics with high school physics vocabulary...

### 5. Cultural phenomenon

**"Problem du millénaire" status** attracted:
- Professional mathematicians (serious attempts)
- Amateurs/hobbyists (enthusiastic but insufficient tools)
- Students testing limits
- Crackpots (less systematic)

**Verification challenge**: Complex proof checking requires expert review. Multiple error points possible.

### 6. Synthesis

**Answer structure**:
1. Accessibility paradox (simple statement, complex proof)
2. Prize incentives (financial motivation)  
3. Scale: thousands submissions
4. Demographics: technical background insufficient
5. Mathematical prerequisites: centuries of development required

**Key insight** !: Not stupidity or lack of rigor - genuine underestimate of mathematical depth needed for solution.

**Physical constraint check**: 100,000 marks substantial sum 1908 → explains competition intensity. Monthly 3-4 submissions realistic given mathematical community size.

**Translation note**: "mauvaises preuves" = bad/incorrect proofs. French mathematical terminology standard.

∴ **Multi-factor explanation**: Accessible statement + substantial prize + verification delays + underestimation of prerequisite mathematics + cultural phenomenon of "impossible problem" attracting attention.

### Answer###
Il y a effectivement eu une grande quantité de preuves incorrectes pour le dernier théorème de Fermat, principalement en raison de la nature même du problème et des enjeux qui l'accompagnaient.

Le théorème de Fermat, qui affirme qu'il n'existe pas de solutions entières positives à l'équation aⁿ + bⁿ = cⁿ pour n supérieur à 2, a attiré des milliers de mathématiciens amateurs et professionnels à sa résolution au fil des siècles. Cette fascination s'explique en partie par le fait que le théorème soit si simple à comprendre mais incroyablement difficile à démontrer.

Les compétitions de prix ont également joué un rôle important. Dès 1816, puis en 1850, l'Académie des Sciences française a proposé des prix pour une preuve générale du théorème. Plus significativement, en 1908, l'industriel allemand Paul Wolfskehl a légué 100 000 marks d'or à l'Académie des Sciences de Göttingen pour récompenser une preuve complète. Cette somme considérable a attiré des candidats de tous horizons, notamment des personnes ayant une formation technique mais une carrière professionnelle défaillante.

Les preuves incorrectes se sont accumulées : en 1907-1908 seule, 621 preuves ont été soumises, et même dans les années 1970, le rythme était encore de 3 à 4 tentatives par mois. Au total, ces échecs ont produit environ 10 pieds de correspondance, témoignant de l'ampleur du phénomène. La plupart de ces preuves se fondaient sur des méthodes élémentaires enseignées dans les écoles, ce qui illustre la difficulté à appréhender pleinement la complexité mathématique nécessaire à sa résolution.