Mixed Sums
of
Primes
and
Other Terms


Initial day:
Feb. 2, 2009
Last modified:
20101122










I. Mixed Sums of Primes and Polygonal Numbers
Conjecture on Sums of Primes and Triangular Numbers (ZhiWei Sun, 2008). (i) Each natural number n ≠ 216 can be written in the form p+T_{x}, where p is a prime or zero, and T_{x}=x(x+1)/2 is a triangular number with x nonnegative. (ii) Any odd integer greater than 3 can be written in the form p+x(x+1), where p is an odd prime and x is a positive integer. Remark. Asymptotically the nth prime is about n*log(n), so the conjecture looks harder than the famous Goldbach conjecture. Parts (i) and (ii) have been verified up to 10^{12} by T. D. Noe and D. S. McNeil respectively. Sun would like to offer 1000 US dollars for the first positive solution or $200 for the first explicit counterexample. General Conjecture on Sums of Primes and Triangular Numbers (ZhiWei Sun, 2008). Let a and b be any nonnegative integers, and let r be an odd integer. Then all sufficiently large integers can be written in the form 2^{a}p+T_{x}, where p is either zero or a prime congruent to r mod 2^{b}, and x is an integer. Also, all sufficiently large odd numbers can be written in the form p+x(x+1), where p is a prime congruent to r mod 2^{b}, and x is an integer. Examples for the General Conjecture on Sums of Primes and Triangular Numbers: (1) (ZhiWei Sun, 2008) Any integer n>88956 can be written in the form p+T_{x}, where p is either zero or a prime congruent to 1 mod 4, and x is a positive integer. (2) (ZhiWei Sun, 2008) Each integer n>90441 can be written in the form p+T_{x}, where p is either zero or a prime congruent to 3 mod 4, and x is a positive integer. (3) (ZhiWei Sun, 2008) Except for 30 multiples of three (the largest of which is 49755), odd integers larger than one can be written in the form p+x(x+1), where p is a prime congruent to 1 mod 4, and x is an integer. (4) (ZhiWei Sun, 2008) Except for 15 multiples of three (the largest of which is 5397), odd numbers greater than one can be written in the form p+x(x+1), where p is a prime congruent to 3 mod 4, and x is an integer. (5) For a=0,1,2,... let f(a) denote the largest integer not in the form 2^{a}p+T_{x}, where p is zero or a prime, and x is an integer. In 2008 ZhiWei Sun conjectured that f(0)=216, f(1)=43473 and f(2)=849591. In 2009 D. S. McNeil suggested that f(3)=7151445. Remark. Jing Ma has verified the above (1)(5) up to 10^{11}. Conjecture on Sums of Primes and Squares (ZhiWei Sun, 2009). If a positive integer a is not a square, then all sufficiently large integers relatively prime to a can be written in the form p+ax^{2}, where p is a prime and x is an integer. In particular, any integer greater than one and relatively prime to 6 can be expressed as p+6x^{2} with p a prime and x an integer. Remark. In 1753 Goldbach asked whether any odd integer n>1 is the sum of a prime and twice a square. In 1856 M. A. Stern and his students found counterexamples 5777 and 5993. Conjecture on Sums of Primes and Polygonal Numbers (ZhiWei Sun, 2009). If a positive integer m>4 is not congruent to 2 mod 8, then all sufficiently large odd numbers can be written as the sum of a prime and twice an mgonal number p_{m}(n)=(m2)n(n1)/2+n (n=0,1,2,...). In particular, any odd integer n>1 other than 135, 345, 539 can be represented by p+2p_{5}(x) = p+3x^{2}x with p a prime and x a nonnegative integer. Remark. Let m>2 be an integer. In 1638 Fermat asserted that any natural number can be expressed as the sum of m mgonal numbers; this was finally proved by Cauchy in 1813. In 2009 ZhiWei Sun conjectured that every natural number n can be written as the sum of an (m+1)gonal number, an (m+2)gonal number, an (m+3)gonal number and a number among 0,...,m3; in the case m=3 this has been verfied for n up to 3*10^{7} by QingHu Hou. Related References
II. Mixed Sums of Primes and Recurrences The wellknown Fibonacci numbers are given by
F_{0}=0,
F_{1}=1,
and F_{n+1}
= F_{n}
+ F_{n1} (n=1,2,3,...).
Here is the list of the initial 18 positive Fibonacci numbers:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,
233, 377, 610, 987, 1597, 2584.
The companion of the Fibonacci sequence is the sequence of
Lucas numbers given by
L_{0}=2,
L_{1}=1,
and L_{n+1}
= L_{n}
+ L_{n1} (n=1,2,3,...).
It is easy to see that L_{n}
= F_{n+1}
+ F_{n1} for n=1,2,3,....
Conjecture on Sums of Primes and Fibonacci Numbers (i) (Weak Version) [ZhiWei Sun, Dec. 23, 2008 and March 19, 2009] Each integer n>4 can be expressed as the sum of an odd prime and two or three positive Fibonacci numbers. (ii) (Strong Version) [ZhiWei Sun, Dec. 26, 2008] Each integer n>5 can be expressed as the sum of an odd prime, a positive Fibonacci numbers and twice a positive Fibonacci number. Also, any integer n>4 can be rewritten as the sum of an odd prime, a positive Fibonacci number and a Lucas number. Remark. Since Fibonacci numbers and Lucas numbers grow exponentially, they are much more sparse than prime numbers. Thus the conjecture seems much more difficult than the Goldbach conjecture. It has been verified up to 10^{12} by D. S. McNeil (Univ. of London). Sun would like to offer $5000 for the first positive solution or $250 for the first explicit counterexample to the above conjecture or the conjecture (an earlier version) that any integer n>4 can be expressed as the sum of an odd prime and two positive Fibonacci numbers. Values of the Representation Function r(n) for n=p+F_{s}+2F_{t} with s,t>1 (n=1,...,100000) Values of the Representation Function r(n) for n=p+F_{s}+L_{t} (s>1, and F_{s} or L_{t} is odd) (n=1,...,100000) Values of the Representation Function r(n) for n=p+F_{s}+F_{t} with s,t>1 and F_{s} or F_{t} odd (n=1,...,100000) Values of the Representation Function r(n) for n=p+F_{s}+F_{t}/2 with s,t>1 (n=1,...,100000) The Pell numbers are defined by
P_{0}=0,
P_{1}=1,
and P_{n+1}
= 2P_{n}
+ P_{n1} (n=1,2,3,...).
Here is the list of the initial 18 positive Pell numbers:
1, 2, 5, 12, 29, 70, 169, 408, 985, 2378,
5741, 13860, 33461, 80782, 195025, 470832, 1136689, 2744210.
Conjecture on Sums of Primes and Pell Numbers (ZhiWei Sun, Jan. 10, 2009). Any integer n>5 can be written as the sum of an odd prime, a positve Pell number and twice a positive Pell number. Remark. This has been verified up to 5*10^{13} by D. S. McNeil. Using a variant of B. Poonen's heuristic arguments, on March 10, 2009 Sun predicted that there should be infinitely many positive integers congruent to 120 modulo 210 not of the form p+P_{s}+2*P_{t} with p an odd prime and s,t positive integers. List of n ≤ 100000 in the Form P_{s}+2*P_{t} (s,t=1,2,3,...) Values of the Representation Function r(n) for n=p+P_{s}+2P_{t} with s,t>0 (n=1,...,100000) The Catalan numbers C_{n}=(2n)!/(n!(n+1)!) (n=0,1,2,...) play important roles in combinatorics. Asymptotically C_{n} ∼ 4^{n} / (n^{3/2}π^{1/2}). It is well known that
C_{0}=1
and C_{n+1}
= C_{0}C_{n}
+ C_{1}C_{n1}
+ … + C_{n}C_{0}
for n=0,1,2,....
Here is the list of the initial 15 Catalan numbers:
1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, 16796,
58786, 208012, 742900, 2674440.
Motivated by the conjecture on sums of primes and Fibonacci numbers due to Sun, QingHu Hou and Jiang Zeng made the following conjecture during their visit to Nanjing Univ. Conjecture (QingHu Hou and Jiang Zeng, Jan. 9, 2009). Every integer n>4 can be expressed as the sum of an odd prime, a positve Fibonacci number and a Catalan number. Remark. Note that the generating function of the Cataln numbers is not rational. The conjecture has been verified up to 3*10^{13} by D. S. McNeil. Hou and Zeng would like to offer $1000 for the first positive solution or $200 for the first explicit counterexample. Values of the Representation Function r(n) for n=p+F_{s}+C_{t} with s>1 and t>0 (n=1,...,100000) The following conjecture is similar to the HouZeng conjecture. Conjecture (ZhiWei Sun, Jan. 16, 2009). Any integer n>4 can be written as the sum of an odd prime, a Lucas number and a Catalan number. Remark. This has been verified up to 10^{13} by D. S. McNeil. Values of the Representation Function r(n) for n=p+L_{s}+C_{t} with t>0 (n=1,...,100000) Related References
III. Mixed Sums of Primes and Powers of Two Conjecture on Sums of Primes and Powers of Two (i) (Weak Version) [ZhiWei Sun, Dec. 23, 2008]. Any odd integer larger than 8 cna be expressed as the sum of an odd prime and three positive powers of two. (ii) (Strong Version) [ZhiWei Sun, Jan. 21, 2009] Each odd number greater than 10 can be written in the form p+2^{x}+3*2^{y} =p+2^{x}+2^{y}+2^{y+1}, where p is an odd prime, and x and y are positive integers. Remark. On Sun's request, Charles Greathouse (USA) verified the conjecture for odd numbers below 10^{10}. It is known that there are infinitely many positive odd integers none of which is the sum of a prime and two powers of 2 (R. Crocker, 1971). Paul Erdos ever asked whether there is a positive integer r such that each odd number greater than 3 can be written as the sum of an odd prime and at most r positive powers of 2. Values of the Representation Function r(n) for 2n1=p+2^{x}+3*2^{y} (n=1,...,50000) Here is Charles Greathouse's list (Feb. 5, 2009) of positive odd integers below 4.45*10^{13} not in the form p+2^{x}+2^{y}. The data above 2.5*10^{11} were obtained subject to his following conjecture. Conjecture (Charles Greathouse, 2009). For an odd integer greater than 5, if it cannot be written as the sum of an odd prime and two positive powers of two, then it must be a multiple of 3*5*17=255. Remark. Greathouse's conjecture implies that any odd number n>8 can be written in the form p+2^{x}+2^{y}+2^{z} with z equal to 1 or 2 since n2 or n4 is not divisible by 3. This gives another strong version for Sun's first assertion in the conjecture on sums of primes and powers of two. D. S. McNeil has verified Greathouse's conjecture for odd numbers below 10^{13}. Open Problem on Sums of Primes and Powers of Two (i) (ZhiWei Sun, Jan. 21, 2009) Prove or disprove that for each k=3,5,...,45,49,51,...,61, any odd integer n>2k+3 can be written in the form p+2^{x}+k*2^{y} with the only exception k=51 and n=353, where p is an odd prime, and x and y are positive integers. (ii) (ZhiWei Sun, Feb. 27, 2009) Prove or disprove the following 47 conjecture: There are infinitely many odd integers greater than 100 and not of the form p+2^{x}+47*2^{y} with p an odd prime and x,y positive integers; moreover, such an odd number is always a multiple of 3*5*7*13=1365. Remark. ZhiWei Sun verified the assertion in part (i) for odd numbers below 2*10^{8}. D. S. McNeil continued the verification for odd numbers below 10^{12} and found no counterexample. 22537515 is the first odd number greater than 100 not of the form p+2^{x}+47*2^{y} (QingHu Hou). Based on D. S. McNeil's search for odd numbers not of the form p+2^{x}+47*2^{y}, ZhiWei Sun formulated the 47 conjecture on Feb. 27, 2009. On Feb. 28, D. S. McNeil yielded a complete list of odd numbers below 10^{13} not of the form p+2^{x}+47*2^{y}, and Sun checked the data and found no counterexample to the 47 conjecture. For odd integer k>61, the number 2k+127 is not of the form p+2^{x}+k*2^{y}. Values of the Representation Function r(n) for 2n1=p+2^{x}+5*2^{y} (n=1,...,50000) Values of the Representation Function r(n) for 2n1=p+2^{x}+11*2^{y} with p ≡ 1 (mod 6) (n=1,...,200000) Values of the Representation Function r(n) for 2n1=p+2^{x}+11*2^{y} with p ≡ 5 (mod 6) (n=1,...,200000) Values of the Representation Function r(n) for 2(n+50)1=p+2^{x}+51*2^{y} (n=1,...,200000) D. S. McNeil's List (Feb. 28, 2009) of Odd Integers in the Interval (100,10^{13}) not of the Form p+2^{x}+47*2^{y} List of n ≤ 100000 in the Form 2^{a}+k*2^{b} (a,b=0,1,2,...) with k=1,3,...,61 In March, 2009, Bjorn Poonen (MIT) used his heuristic arguments to make the following prediction based on the Generalized Riemann Hypothesis (GRH). Poonen's Prediction (March 6, 2009). For each positive integer k, any infinite arithmetic progression of positive odd integers contains infinitely many integers not of the form p+2^{x}+k*2^{y}, where p is an odd prime, and x and y are positive integers. Related References




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