Let $T(n^k)$ be the threshold of completeness of the $k$-th powers, the largest number which is not the sum of distinct $k$-th powers. Ërdos and Graham conjectured in 1980 that the sequence of these thresholds is not eventuallly monotonic, in other words, $T(n^k)>T(n^{k+1})$ happens infinitely often. But, it turns out that the sequence might be monotonic after all. In a paper by Kim (Kim, D. On the largest integer that is not a sum of distinct positive nth powers, Journal of Integer Sequences, Volume 20, Issue 7 (2017)), upper bounds for these thresholds are given, but these are very far from the lower bounds, meaning that even when using good candidates for $k$ that might break the monotonicity, such as $k=8$, the very next threshold might be closer to the upper bound that the lower bound. In fact computations by Michael J. Wiener in the paper "The Largest Integer Not the Sum of Distinct 8th Powers" have shown that $T(n^8)<T(n^9)$. This is an interesting case of a conjecture that was genuinely believed, despite plenty of computational evidence pointing in the opposite direction. But it could still be true. It is very outrageous, and It has some importance for the additive theory of $k$-th powers and complete sequences.

Let $T(n^k)$ be the threshold of completeness of the $k$-th powers, the largest number which is not the sum of distinct $k$-th powers. Ërdos and Graham conjectured in 1980 that the sequence of these thresholds is not eventuallly monotonic, In other words, $T(n^k)>T(n^{k+1})$ happens infinitely often. But, it turns out that the sequence might be monotonic after all. In a paper by Kim (Kim, D. On the largest integer that is not a sum of distinct positive nth powers, Journal of Integer Sequences, Volume 20, Issue 7 (2017)), upper bounds for these thresholds are given, but these are very far from the lower bounds, meaning that even when using good candidates for $k$ that might break the monotonicity, such as $k=8$, the very next threshold might be closer to the upper bound that the lower bound. In fact computations by Michael J. Wiener in the paper "The Largest Integer Not the Sum of Distinct 8th Powers" have shown that $T(n^8)<T(n^9)$. This is an interesting case of a conjecture that was genuinely believed, despite plenty of computational evidence pointing in the opposite direction. But it could still be true. It is very outrageous, and It has some importance for the additive theory of $k$-th powers and complete sequences.

Let $T(n^k)$ be the threshold of completeness of the $k$-th powers. Ërdos and Graham conjectured in 1980 that the sequence of these thresholds is not eventuallly monotonic, in other words, $T(n^k)>T(n^{k+1})$ happens infinitely often. But, it turns out that the sequence might be monotonic after all. In a paper by Kim (Kim, D. On the largest integer that is not a sum of distinct positive nth powers, Journal of Integer Sequences, Volume 20, Issue 7 (2017)), upper bounds for these thresholds are given, but these are very far from the lower bounds, meaning that even when using good candidates for $k$ that might break the monotonicity, such as $k=8$, the very next threshold might be closer to the upper bound that the lower bound. In fact computations by Michael J. Wiener in the paper "The Largest Integer Not the Sum of Distinct 8th Powers" have shown that $T(n^8)<T(n^9)$. This is an interesting case of a conjecture that was genuinely believed, despite plenty of computational evidence pointing in the opposite direction. But it could still be true. It is very outrageous, and It has some importance for the additive theory of $k$-th powers and complete sequences.

Let $T(n^k)$ be the threshold of completeness of the $k$-th powers, the largest number which is not the sum of distinct $k$-th powers. Ërdos and Graham conjectured in 1980 that the sequence of these thresholds is not eventuallly monotonic, in other words, $T(n^k)>T(n^{k+1})$ happens infinitely often. But, it turns out that the sequence might be monotonic after all. In a paper by Kim (Kim, D. On the largest integer that is not a sum of distinct positive nth powers, Journal of Integer Sequences, Volume 20, Issue 7 (2017)), upper bounds for these thresholds are given, but these are very far from the lower bounds, meaning that even when using good candidates for $k$ that might break the monotonicity, such as $k=8$, the very next threshold might be closer to the upper bound that the lower bound. In fact computations by Michael J. Wiener in the paper "The Largest Integer Not the Sum of Distinct 8th Powers" have shown that $T(n^8)<T(n^9)$. This is an interesting case of a conjecture that was genuinely believed, despite plenty of computational evidence pointing in the opposite direction. But it could still be true. It is very outrageous, and It has some importance for the additive theory of $k$-th powers and complete sequences.