Summary
In this paper, we study mixed sums of primes and linear recurrences. We show that if m ≡ 2 (mod 4) and m + 1 is a prime, then \(({m}^{{2}^{n}-1 } - 1)/(m - 1)\not ={m}^{n} + {p}^{a}\) for any n = 3, 4, … and prime power p a. We also prove that if a > 1 is an integer, u 0 = 0, u 1 = 1, and \({u}_{i+1} = a{u}_{i} + {u}_{i-1}\) for i = 1, 2, 3, …, then all the sums \({u}_{m} + a{u}_{n}\ (m,n = 1,2,3,\ldots )\) are distinct. One of our conjectures states that any integer n > 4 can be written as the sum of an odd prime and two positive Fibonacci numbers.
In honor of Prof. M.B. Nathanson on the occasion of his 60th birthday
Mathematics Subject Classifications (2000). Primary 11P32, Secondary 11A41, 11B37, 11B39, 11B75, 11Y99
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References
J.-R. Chen, On the representation of a large even integer as the sum of a prime and the product of at most two primes, Sci. Sinica 16 (1973), 157–176
F. Cohen and J. L. Selfridge, Not every number is the sum or difference of two prime powers, Math. Comp. 29 (1975), 79–81
R. Crocker, On a sum of a prime and two powers of two, Pacific J. Math. 36 (1971), 103–107
P. Erdős, On integers of the form 2 k + p and some related problems, Summa Brasil. Math. 2 (1950), 113–123
P. Erdős, Some of my favorite problems and results, in: The Mathematics of Paul Erdős, I (R. L. Graham and J. Nešetřil, eds.), Algorithms and Combinatorics 13, Springer, Berlin, 1997, pp. 47–67
P. X. Gallagher, Primes and powers of 2, Invent. Math. 29 (1975), 125–142
R. K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, New York, 2004
D. R. Heath-Brown and J.-C. Puchta, Integers represented as a sum of primes and powers of two, Asian J. Math. 6 (2002), 535–565
Q. H. Hou and J. Zeng, Sequences A154404 in On-Line Encyclopedia of Integer Sequences, http://www.research.att.com/~njas/sequences/A154404
Yu. V. Linnik, Prime numbers and powers of two, Trudy Mat. Inst. Steklov. 38 (1951), 152–169
D. McNeil, Sun’s strong conjecture (a message to Number Theory Mailing List in Dec. 2008), http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0812\&L=nmbrthry\&T=0\&P=3020
D. McNeil, Various and sundry (a message to Number Theory Mailing List in Jan. 2009), http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0901\&L=nmbrthry\&T=0\&P=840
D. McNeil, Personal communications in January 2009
M. B. Nathanson, Additive Number Theory: The Classical Bases, Grad. Texts in Math., Vol. 164, Springer, New York, 1996
J. Pintz and I. Z. Ruzsa, On Linnik’s appproximation to Goldbach’s problem, I, Acta Arith. 109 (2003), 169–194
R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999
Z. W. Sun, On integers not of the form ± p a ± q b, Proc. Amer. Math. Soc. 128 (2000), 997–1002
Z. W. Sun, On sums of primes and triangular numbers, Journal of Combinatorics and Number Theory 1 (2009), 65–76
Z. W. Sun, Three messages to the Number Theory Mailing List,http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0812\&L=nmbrthry\&T=0\&P=2140 http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0812\&L=nmbrthry\&T=0\&P=2704 http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0812\&L=nmbrthry\&T=0\&P=3124
Z. W. Sun, Sequences A154257, A154258, A154263, A154536 in On-Line Encyclopedia of Integer Sequences,http://www.research.att.com/~njas/sequences/A154257 http://www.research.att.com/~njas/sequences/A154258 http://www.research.att.com/~njas/sequences/A154263 http://www.research.att.com/~njas/sequences/A154536
Z. W. Sun, Sequences A154285, A154417, A154290, A154421 in On-Line Encyclopedia of Integer Sequences,http://www.research.att.com/~njas/sequences/A154285 http://www.research.att.com/~njas/sequences/A154290 http://www.research.att.com/~njas/sequences/A154417
Z. W. Sun, Offer prizes for solutions to my main conjectures involving primes (a message to the Number Theory Mailing List in January 2009), http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0901\&L=nmbrthry\&T=0\&P=1395
Z. W. Sun and M. H. Le, Integers not of the form \(c({2}^{a} + {2}^{b}) + {p}^{\alpha }\), Acta Arith. 99 (2001), 183–190
K. J. Wu and Z. W. Sun, Covers of the integers with odd moduli and their applications to the forms x m − 2 n and \({x}^{2} - {F}_{3n}/2\), Math. Comp., 78 (2009), 1853–1866
G. Zeller-Meier, Not prime for each n ≥ 3 (a message to Number Theory Mailing List in March 2005), http://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0503\&L=nmbrthry\&T=0\&P=2173
Acknowledgements
Research supported by the National Natural Science Foundation of China (grant 10871087). The author wishes to thank Dr. Douglas McNeil who has checked almost all conjectures mentioned in this paper via his quite efficient and powerful computation.
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Sun, ZW. (2010). Mixed Sums of Primes and Other Terms. In: Chudnovsky, D., Chudnovsky, G. (eds) Additive Number Theory. Springer, New York, NY. https://doi.org/10.1007/978-0-387-68361-4_24
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DOI: https://doi.org/10.1007/978-0-387-68361-4_24
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