Research on density estimates for exceptional sets associated with additive representations of natural numbers
| Project/Area Number |
17540004
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Iwate University |
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Principal Investigator |
KAWADA Koichi Iwate University, Faculty of Education, Dept of Math, Associate Professor (70271830)
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| Co-Investigator(Kenkyū-buntansha) |
OSHIKIRI Gen-ichi Faculty of Education, Dept of Math, Professor (70133931)
NAKAJIMA Fumio Faculty of Education, Dept of Math, Professor (20004484)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,830,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥330,000)
Fiscal Year 2007: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2006: ¥1,100,000 (Direct Cost: ¥1,100,000)
Fiscal Year 2005: ¥1,300,000 (Direct Cost: ¥1,300,000)
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| Keywords | Waring''s Problem / Goldbach''s problem / Powers of Integers / Cubes / Biquadrates / Prime numbers / Almost Primes / Waring問題 / 漸近式 / Goldbach問題 / 滑らかな数 |
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| Research Abstract |
The most central accomplishment of this three year research project is that we succeeded in determining completely the integers that cannot be written as the sum of at most sixteen biquadrates. We in particular contributed to the proof that every integer exceeding 216th power of 10 and not divisible by 16 can be written as the sum of sixteen biquadrates, by making use of the circle method. We also determined explicitly the exceptional sets associated with the sum of seventeen and eighteen biquadrates. For example, we established that every natural number can be written as the sum of eighteen biquadrates except only for the following seven numbers; 79, 159, 239, 319, 399, 479 and 559. As a by-product, moreover, we provided a new proof, substantially simpler than the previous one, of the fact that every natural number can be written as the sum of at most nineteen biquadrates.
Next, concerning the Waring-Goldbach problem, we showed that every sufficiently large integer can be written as the sum of seven cubes of natural numbers that have at most four prime factors counted according to multiplicity. This conclusion improves the previously best result in this direction, in which the upper limit for the number of prime factors was 69 in place of four. The latter theorem was proved by applying sieve methods and the circle method.
Besides these two main products mentioned above, we worked on the exceptional sets associated with the following additive problems; sums of seven cubes of smooth numbers (that means natural numbers having smaller primes factors only), sums of fifteen biquadrates, and sums of five cubes of primes. We intend to continue our research in this area.
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Report
(4 results)
Research Products
(36 results)
