A currently true statement \Gamma of the form "\exists f:{\mathbb N} \to {\mathbb N} of unknown computability ...", where f is conjecturally uncomputable, \Gamma remains unproven when f is computable, and \Gamma holds forever when f is uncomputable

Agnieszka Peszek and Apoloniusz Tyszka,
On the Relationship Between Matiyasevich's and Smorynski's Theorems,
Sci. Ann. Comp. Sci. 29 (2019), no. 1, 101-111.

A hypothetical upper bound on the heights of the solutions of a Diophantine equation with a finite number of solutions,
Open Comput. Sci. 8 (2018), no. 1, 109-114.

Is there a computable upper bound for the height of a solution of a Diophantine equation with a unique solution in positive integers?
Open Comput. Sci. 7 (2017), no. 1, 17-23.

Krzysztof Molenda, Agnieszka Peszek, Maciej Sporysz, Apoloniusz Tyszka,
Is there a computable upper bound on the heights of rational solutions of a Diophantine equation with a finite number of solutions?
Proceedings of the 2017 Federated Conference on Computer Science and Information Systems, M. Ganzha, L. Maciaszek, M. Paprzycki (eds). ACSIS, vol. 11, pp. 249–258 (2017).

A new characterization of computable functions,
An. Stiint. Univ. Ovidius Constanta Ser. Mat. 21 (2013), no. 3, pp. 289-293.

Does there exist an algorithm which to each Diophantine equation assigns an integer
which is greater than the modulus of integer solutions, if these solutions form a finite set?
Fundamenta Informaticae, vol. 125, no. 1, pp. 95–99, 2013.

Conjecturally computable functions which unconditionally do not have any finite-fold Diophantine representation,
Information Processing Letters, vol. 113, no. 19-21, pp. 719–722, 2013.

Two conjectures on the arithmetic in R and C,
Mathematical Logic Quarterly, 56 (2010), no.2, pp.175-184.

Some conjectures on addition and multiplication of complex (real) numbers,
International Mathematical Forum 4 (2009), no.9-12, pp.521-530.

Andrzej Kozlowski and Apoloniusz Tyszka,
A Conjecture of Apoloniusz Tyszka on the Addition of Rational Numbers, The Wolfram Demonstrations Project.

Arithmetic neighbourhoods of numbers,
International Mathematical Forum 3 (2008), no.5-8, pp.349-362.

Arithmetic neighbourhoods of numbers, an extended abstract.

A system of equations,
SIAM Problems and Solutions (electronic only), Problem 07-006, 2007.

Steinhaus' problem on partition of a triangle,
Forum Geometricorum 7 (2007), pp.181-185.

On $\emptyset$-definable elements in a field,
Collectanea Mathematica 58 (2007), no.1, pp.73-84.

A discrete form of the Beckman-Quarles theorem for mappings from R^2 (C^2) to F^2, where F is a subfield of a commutative field extending R (C),
J. Geom. 85 (2006), no.1-2, pp.188-199.

The Beckman-Quarles theorem for continuous mappings from C^n to C^n,
Aequationes Math. 72 (2006), no.1-2, pp.78-88.

A discrete form of the theorem that each field endomorphism of R (Q_p) is the identity,
Aequationes Math. 71 (2006), no.1-2, pp.100-108.

The Beckman-Quarles theorem for mappings from C^2 to C^2,
Acta Math. Acad. Paedagog. Nyházi. (N.S.) 21 (2005), no.1, pp.63-69 (electronic), www.emis.de/journals

The Beckman-Quarles theorem for mappings from R^2 to F^2, where F is a subfield of a commutative field extending R,
Abh. Math. Semin. Univ. Hamb. 74 (2004), pp.77-87.

Beckman-Quarles type theorems for mappings from R^n to C^n,
Aequationes Math. 67 (2004), pp.225-235.

A stronger form of the theorem on the existence of a rigid binary relation on any set,
Aequationes Math. 66 (2003), pp.257-260.

A discrete form of the Beckman-Quarles theorem for two-dimensional strictly convex normed spaces,
Nonlinear Funct. Anal. Appl. 7 (2002), no.3, pp.353-360.

On binary relations without non-identical endomorphisms,
Aequationes Math. 63 (2002), pp.152-157.

Discrete versions of the Beckman-Quarles theorem from the definability results of R. M. Robinson,
Algebra, Geometry & their Applications, Seminar Proceedings 1 (2001), pp.88-90.

A discrete form of the Beckman-Quarles theoem for rational eight-space,
Aequationes Math. 62 (2001), pp.85-93.

A combinatorial characterization of second category subsets X^\omega,
J. Nat. Geom. 18 (2000), no.1-2, pp.125-130.

Discrete versions of the Beckman-Quarles theorem,
Aequationes Math. 59 (2000), pp.124-133.

On the minimal cardinality of a subset of R which is not of first category,
J. Nat. Geom. 17 (2000), no.1-2, pp.21-28.

A discrete form of the Beckman-Quarles theorem,
Amer. Math. Monthly, 104 (1997), no.8, pp.757-761.

Subobjects of the successive power objects in the topos G-Set,
J. Nat. Geom. 11 (1997), no.2, pp.129-130.

On a combinatorial property of families of sequences converging to \infty,
J. Nat. Geom. 11 (1997), no.1, pp.51-56.

On combinatorial problem concerning partitions of a box into boxes,
J. Nat. Geom. 8 (1995), no.2, pp.129-132, full text.

On combinatorial problem concerning partitions of a box into boxes,
J. Nat. Geom. 8 (1995), no.2, pp.129-132, Zbl review.

On some combinatorial problems concerning partitions of a box,
J. Nat. Geom. 5 (1994), no.1, pp.1-9, full text.

On some combinatorial problems concerning partitions of a box,
J. Nat. Geom. 5 (1994), no.1, pp.1-9, Zbl review.