Problems & Puzzles: Conjectures

Conjecture 55. n = m^2 + p + d

Werner Sand sends the following conjecture:

Each natural number n > 2 is the sum of a square (>0) and a prime or the sum of a square, a prime, and 1.

n = m^2 + p + d, d = 0 v 1.

We can formulate the Goldbach Conjecture (even version) in this way: Each
natural number n > 3 is the sum of 2 primes and (0 or 1). This has the same
structure as my conjecture (except for the fact that the "1" is trivial).
But my conjecture is much stronger than the GC, because there are less
squares in an interval than primes (less density). E.g.: For n=10^6 there
are more than 5.000 Goldbach partitions but only 63 "Sand partitions". The
conjecture has also a certain similarity to the four squares theorem (each
positive integer is the sum of at most 4 squares), because p can be the sum
of 2 squares, but it is stronger.

With growing values of n, the difference d=1 is becoming relatively
unimportant and increasingly rare, if standing alone (no d=0 for the same
n). So we can say (short form of the conjecture): Each natural number n>2 is exactly or nearly the sum of a square and a prime.

If you concentrate only in n=m^2+p, m>0, then obviously every n=x^2>9 has no solution, except when n-m=1 & n+m is prime.

What about solutions for non-square n values? The following 35 non-square n values has no solution type n=m^2+p:

5 10 13 31 34 37 58 61 85 91 127 130 214 226 370 379 439 526 571 706 730 771 829 991 1255 1351 1414 1549 1906 2986 3319 3676 7549 9634 21679

As a matter of fact Crespi de Valldaura conjectures that 21679=7*19*163, is "Almost certainly the largest non-square integer which is not the sum of a square and a prime"

Q2. Is there a largest numbers such as 21679?

Contributions came from Farideh Firoozbakht & Patrick Capelle. Capelle's contribution is a small treatise of the current status of the art of the issue, plenty of references.

***

Farideh wrote:

It seems that in the following statement,

" If you concentrate only in n=m^2+p, m>0, then obviously every n=x^2>9 has
no solution,  except when n-m=1 & n+m is prime."

you made a typo mistake. The correct statement is:

"..., except when x-m=1 & x+m is prime." namely "..., when 2*n^(1/2)-1=2x-1 is prime.".

Answer to Q1: I think this conjecture is true. I checked it for all square numbers up to
10^12 and for all non-square numbers up to 10^7.

Answer to Q2: I think the Crespi de Valldaura's conjecture is true.
I checked the validity of this conjecture up to n=10^7.

We can generalize this conjecture as follows:

For each natural number k, k>1 there exists a nutural number f(k) such
that f(k) isn't of the form m^k & m^k+p where m>0 and p is a prime and
if n>f(k) and n isn't of the form n=m^k then n is of the form m^k+p .

For example Crespi de Valldaura's conjecture says f(2)=21679 and I guess that
f(3)=78526384.

***

Patrick wrote:

Many authors have been interested by the numbers which are the sum of a square and a prime, and some of them (Klaus Brockhaus, Richard K. Guy, Axel Harvey, Dean Hickerson, Vladeta Jovovic, T.D. Noe, Mike Oakes, John Robertson, Felice Russo, Robert G. Wilson, Allessandro Zacagnini, Reinhard Zumkeller and others) have worked on some particular cases (e.g., n prime) or generalizations (e.g., replacement of the square m^2 by a power)  ,,,,,. The conjecture proposed by Werner Sand is interesting and probably new. It held my attention by its simplicity and the use of the logical symbol of disjunction. It is difficult to understand the origin of this disjunction (in case it corresponds to the association of several conjectures/ideas). So, for this analysis, starting with some observations already known (there are an infinity of squares n = a^2 which are not the sum of a square and a prime  ; there are an infinity of numbers which are not the sum of a nonzero square and a prime  ; see also the table of Noe , including the CR list), i collected several ideas coming from the net to transform them into conjectures (some of them are not new, except their formulation) and comments.

Conjecture A :
Every natural number n > 21679 is a square or the sum of a square and a prime number  ,.
In other words, every natural number n > 21679 is a square, a prime number or the sum of a nonzero square and a prime number.
It's an improvement of a well known conjecture of Hardy and Littlewood (conjecture H : every large number is either a square or the sum of a square and a prime) .
James Van Buskirk checked it up to n = 3000000000  ,,.

Conjecture B :
Every natural number n > 21679 is a square or the sum of a nonzero square and a prime number.
Note that the conjecture A becomes the conjecture B by using the conjecture D (when p > 21679).

Conjecture C :
Every nonsquare number n > 21679 is the sum of a nonzero square and a prime number.
Note that the conjecture C is equivalent to the conjecture B.

Conjecture D :
Every prime number p > 7549 is the sum of a nonzero square and a prime number  .
It's an improvement of a conjecture of Mike Oakes (conjecture C in  ).

Conjecture E :
Every natural number n > 21679, different from a square a^2 such that 2a - 1 is composite, is the sum of nonzero square and a prime number.

Conjecture F :
Every square number a^2, such that 2a - 1 is composite, can be expressed as a^2 = m^2 + p + 1, with m > 0 and p prime.

Conjecture G :
Every square number a^2, such that 2a - 1 is prime, can be expressed as a^2 = c^2 + p = d^2 + q + 1, with c > 0, d > 0, p prime, q prime.
It means that if a square is the sum of a nonzero square and a prime number, then it is the sum of a nonzero square, a prime number and 1.

Conjecture H :
Every square number a^2 > 1 can be expressed as a^2 = m^2 + p + 1, m > 0, p prime.

The conjecture C does not mean that there is no square which is sum of a nonzero square and a prime. Among the values of n, we have in fact two kinds of squares : the squares n = a^2 where 2a - 1 is prime (squares that i call R) and the squares n = a^2 where 2a - 1 is composite (squares that i call S). It's a consequence of Hickerson's proof  : if a^2 = m^2 + p, with p prime, then p = (a-m)(a+m) ; so a - m = 1 and a + m = p ; hence 2a -1 = p is prime. If 2a - 1 is not prime, then two possibilities : 2a - 1 = 1 (so a = 1 and n = 1, but it cannot be accepted because Werner Sand specified that n > 2) or 2a - 1 is composite. In this last case, n = a^2 cannot be expressed as the sum of a (nonzero) square and a prime number. A square R is always the sum of a nonzero square and a prime, but it's never the case for a square S. The conjecture E is a consequence of this situation. Considering the conjectures above, I suppose that Werner Sand added the possibility d = 1 essentially for cases where n is a square S. If the main idea behind his conjecture is true, then i guess that every square S can be expressed as S = m^2 + p + 1 (conjecture F). But it doesn't imply that the squares R don't have the same property ! In fact, the squares R can also be expressed in the form m^2 + p + 1, which gives me opportunity to propose the conjecture G. Hence, in the formulation of Sand's conjecture, d = 0 or 1 should be replaced by d = 0 and/or 1, because there are numbers n for which d = 0 and d = 1. The conjecture H is a consequence of the conjectures F and G. Are there finitely many nonsquare numbers which cannot be expressed in the form m^2 + p + 1, m > 0, p prime ? I wonder whether every sufficiently large number can be expressed in this form.

References :
 Integers of the form x^2+kp, by J. Robertson (Mar 14, 1999 and interesting comments of Mar 20,1999).
 A065376 : Primes of the form p + k^2, p prime and k > 0 (Nov 3, 2001).
 A073770 : Primes p not of the form q + s where q is prime and s > 0 is the smallest square such that q + s is prime (Aug 8, 2002).
 Four conjectures of Mike Oakes (Jun 18, 2006). The conjectures C and D are especially concerned.
 A conjecture regarding the form p = b^m + q , by Mike Oakes (Jun 19, 2006, and reply of Jun 20, 2006).
 A014090 : Numbers that are not the sum of a square and a prime.
 A064233 : Numbers which are not the sum of a prime number and a nonzero square.
 Sequence A020495 (it's the CR list, but without the prime numbers).

***

Werner Sand wrote on Oct 29:

Thanks for Farideh Firoozbakht's generalization and Patrick Capelle's detailed documentation.

I din't know de Valldaura's conjecture (VC) or other investigation in the vicinity of the subject. I came from the following consideration: The sum of at most 4 squares can represent the natural numbers (Lagrange).  This theorem I tried to narrow and I found that 2 squares and 1 prime are sufficient, even if the prime is not the sum of  2 squares (Euler). Finally, I was able to reduce one of the 2 squares to the numbers 0 and 1, and this was my conjecture. The fact that for n=square is always n = k^2 = m^2+p, i.e. p = k^2-m^2 = (k+m)(k-m), thus p can be prime if k-m = 1, was of course my first thought. But p = k+m is not necessarily prime, i.e. a square is not necessarily the sum of a square and a prime, e.g. k(64) = 8, m(64) = 7, k+m = 15. In these cases, however, n-1 is the sum of a square and a prime (d=0, my conjecture), and then of course d(n)=1 (our example: 63 = 4+59 = 16+47). Thus there cannot be 2 consecutive n with d=only 1. It was evident that there are nonsquare n besides the squares which are not the sum of a square and a prime. However, I considered their number to be infinite, because I didn't know the VC. In my opinion, there is little sense to differ square from nonsquare n as long as the VC is not proven. But it seems to me  important that there are infinitely many n with d(n)=0 and infinitely many n with d(n)=1. And of course that there is no need for d>1.

As to the formulations "d = 0 v 1" and "d = 0 or 1": v  comes from Latin vel and is defined as "either…or…or both". It is not disjunction as P. Capelle says, but adjunction. As well in the German mathematical  diction "oder" means  "entweder…oder…oder beide". I guess this applies to the English "or", too. So there is no reason to change the formulation.  By the way I would have preferred set symbols, but unfortunately this is not possible.

Outlook: We can continue the procedure of refining and strengthening and restrict m^2 to p^2, i.e. square primes. A single additional value of d (d=2) allows this:

Conjecture:

Every natural number n>5 is the sum of a square prime, a prime, and one of the values 0,1,2: n = p^2+q+d,  d out of (0,1,2).

I checked this until n=10^6. Comparison number of partitions of n=10^6:

Goldbach: 5.402
m^2+p+d: 63
p^2+q+d: 47

...

In order to complete F. Firoozbakht's generalization:

For each natural number k there is a natural number D such that for each
natural number n>2 applies:

n = m^k+p+d,  d of {0,1,2…D}, m  natural number,  p  prime number.

Some values:

k                      D
1                      0
2                      1
3                      3
4                      8
5                      15
6                      17
7                      22

We can call D a "measure of uncertainty".

***

Farideh Firoozbakht wrote (Nov 15):

Jaroslaw Wroblewski kindly wrote his comment about my generalization
of conjecture 55:

"Looks reasonable to me if k is prime. Otherwise I would change
n=m^k to n=m^d, d being a divisor of k greater than 1."

So the generalized form of conjecture 55 can be true only for primes k,
and we must improve it in the following form.

For each prime k, there exists a nutural number f(k) such that f(k) isn't of
the form m^k & m^k+p where m>0, p is a prime and if n>f(k) and n isn't
of the form n=m^k then n is of the form m^k+p .

***

Richard Chen wrote on March 2022:

This set is the union of A020495 (composites) and A065377 (primes), A020495 is conjectured to be finite with 21 terms (largest term is 21679), A065377 is conjectured to be finite with 15 terms (largest term is 7549), thus this set is also conjectured to be finite, with 21+15=36 terms.

Although this set currently has no OEIS sequences, there is a sequence A064233 in OEIS for all numbers not of the form m^2+p with m>=1 and prime p, including the square numbers, the square numbers in A064233 are A104275^2, thus the sequence A064233 is infinite, but this set (which is the nonsquares in A064233) is conjectured to be finite, also there is a sequence A014090 in OEIS for all numbers not of the form m^2+p with m>=0 and prime p, i.e. 0 is counted as a square (like A020495), since the A014090 also contain A104275^2 as a subsequence of squares, A014090 is also infinite. Besides, there are subsequences of this set in OEIS: A317966 (numbers == 1 or 2 mod 4) and A308516 (odd composites). There is also OEIS sequence for the number of ways to write n as m^2+p with p prime: A064272 (m>=1) and A002471 (m>=0)

A related problem of this problem is for odd numbers n not of the form 2*m^2+p with m>=1 and prime p, which is the sequence A060003, and the primes in A060003 (together with the "oddest" prime 2) are the Stern primes A042978, A060003 is also conjectured to be finite with 10 terms (largest term is 5993). There is also OEIS sequence for the number of ways to write n as 2*m^2+p with p prime: A143539 (m>=1) and A046923 (m>=0). There is a sequence A347567 for the combine with these two problems, the even numbers in A347567 are 2*(the numbers in this set) together the number 6 (6 is the only number requiring the even prime 2), i.e. the union of 2*A020495 and 2*A065377 and {6}, and the odd numbers in A347567 are the numbers in A060003, since all of A020495, A065377, A060003 are conjectured to be finite, with 21, 15, 10 terms, respectively, A347567 is also conjectured to be finite, with 21+15+10+1=47 terms, and the largest term is 2*21679=43358

There is also a similar problem about the triangular numbers (rather than the square numbers), the nontriangular numbers not of the form m*(m+1)/2+p with m>=1 and prime p, this set is A255904, the primes in A255904 are A065397, and it is conjectured that 216 is the only composite in A255904, A065397 is conjectured to be finite with 4 terms (largest term is 211), thus A255904 is also conjectured to be finite, with 4+1=5 terms, and there is a sequence A111908 in OEIS for all numbers not of the form m*(m+1)/2+p with m>=1 and prime p, including the triangular numbers, the triangular numbers in A111908 are A138666*(A138666+1)/2, thus the sequence A111908 is infinite, but this set (which is the nontriangular numbers in A111908) is conjectured to be finite, also there is a sequence A076768 in OEIS for all numbers not of the form m*(m+1)/2+p with m>=0 and prime p, i.e. 0 is counted as a triangular number, since the A076768 also contain A138666*(A138666+1)/2 as a subsequence of squares, A076768 is also infinite. Although there is no OEIS sequence for the number of ways to write n as m*(m+1)/2+p with p prime, A132399 is the sequence for "p is 0 or prime" rather than "p is prime", but of course there is no difference between these two sequences for nontriangular numbers n.

A related problem of this problem is for odd numbers n not of the form m*(m+1)+p with m>=1 and prime p, and it is conjectured that 1 and 3 are the only such odd numbers. There is also OEIS sequence for the number of ways to write n as m*(m+1)+p with p prime: A144590 (m>=1). There is a sequence A347568 for the combine with these two problems, the even numbers in A347568 are 2*A255904 together the number 10 (10 is the only number requiring the even prime 2), i.e. the union of (2*the set of composites in A255904, conjectured to be {432}) and 2*A065397 and {10}, and the odd numbers in A347568 are conjectured to be {1, 3}, since all of these three sequences are conjectured to be finite, with 1, 4, 2 terms, respectively, A347568 is also conjectured to be finite, with 1+4+2+1=8 terms, and the largest term is 2*216=432

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